Points that were vertices before the reflection are still vertices after the reflection (just different vertices) and points that formed straight edges before the reflection still form straight edges after the reflection (just different straight edges).
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Instead of considering a finite number of vertices, as there would be on a map, it considers infinitely many vertices, one for each point in the plane.
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And Kneser had asked, in 1955 or '56, how many colors are required to color all the vertices if vertices that are connected must be different colors.
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You might want to know whether it's possible to color the vertices using three colors, so that no vertices connected by an edge have the same color.
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Finally, the polyhedron has 30 vertices where 4 triangles meet.
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Their 120-sided polyhedron has 12 vertices where 10 triangles meet.
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In addition, the polyhedron has 20 vertices where 6 triangles meet.
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The kiosk is cuboid but with sinuous curves instead of vertices.
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Imagine a graph—a collection of dots (vertices) connected by lines (edges).
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The verifier wants the provers to report the colors of connected vertices.
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The vertices A1, A2, …, Ak have integral coordinates and lie on a circle.
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Andy Rolfes: No, because you had to move vertices around to make swords.
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This says that if you have a planar graph (a network of vertices and edges in the plane) that stays connected if you remove one or two vertices, then there is a convex polyhedron that has exactly the same connectivity pattern.
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Mathematicians are interested in the following: How many different ways can you color the vertices of the triangle, given some number of colors and adhering to the rule that whenever two vertices are connected by an edge, they can't be the same color.
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Examples of vertices that do well are comedy, memes, dance, vlogs, creation/DIY and hacks.
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To help us identify specific symmetries, let's start by labeling the vertices of the original square.
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Maxwell developed a theoretical color space, a triangle with red, green, and blue at the vertices.
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Think of it as upping the number vertices in the model of a player's personal style.
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Those probabilities are wired into the structure as "weights" and directions in the lines between the vertices.
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Others rise and fall in a tessellation of triangles with sharp, jewel-like edges, their vertices multiplying.
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So instead, you could ask each prover to tell you the color of one of two connected vertices.
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On Saturday, Marijn Heule, a computer scientist at the University of Texas, Austin, found one with just 874 vertices.
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Feynman diagrams have a basic geometric aspect to them, formed as they are from line segments, rays and vertices.
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They are situated at the eight vertices of a large cube (and so 4 of them are above ground).
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The objects are shown as points (or vertices) and the pair relationship is shown with an arrow (a directed edge).
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You can imagine an infinite number of graphs—graphs with more vertices and more edges connected in any number of ways.
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She and colleagues used a two-dimensional model of cells that are connected along edges and at vertices, filling all space.
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Soon enough, Dustin Mixon, a mathematician at Ohio State University, and his collaborator Boris Alexeev found a graph with 1,577 vertices.
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If the vertices aren't connected, then the answers to the questions won't say anything about whether the graph is three-colored.
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In other words, the image of a side of the square is determined by the image of the vertices that are its endpoints.
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But even this interrogation strategy fails as graphs get really big—with more edges and vertices than there are atoms in the universe.
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He was later able to shrink the graph to 1,581 vertices and do a computer check to verify that it was not four-colorable.
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The hope is that the two vertices are connected to each other, even though neither prover knows which vertex the other is thinking about.
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In the Thanksgiving passenger graph, the directed graph shows the increase in traffic for Thanksgiving between flight origins and destinations, which are the vertices.
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In a sphere, the curvature is distributed evenly over the entire surface; in a cube, it's concentrated in equal amounts at the eight evenly spaced vertices.
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These tapered female forms seem semi-linked, their long limbs and hyper-extended torsos forming lattice-like vertices behind which a foregrounded pair of oversized eyes stare.
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The robot lowers its hands, and the cubes coalesce into a single shape—with 24 square faces, 16 vertices, and eight connected cubes existing in four dimensions.
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Nelson asked: What is the smallest number of colors that you'd need to color any such graph, even one formed by linking an infinite number of vertices?
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To find a lower bound for the chromatic number, it suffices to create a graph with a finite number of vertices that requires a particular number of colors.
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The problem is easy to understand and start working on, and there is a clear measure of success: lowering the number of vertices in a non-four-colorable graph.
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At the time, he felt stymied by a mathematical problem involving simplexes—a simplex is the polygon with the fewest vertices in any given dimension—and he wanted a break.
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Any of the Steinitz-type proofs will tell you not only that there is a polyhedron but also that there's a polyhedron with integers for the coordinates of the vertices.
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It is impossible to partition the Schönhardt polyhedron into tetrahedra whose vertices are vertices of the polyhedron. More strongly, there is no tetrahedron that lies entirely inside the Schönhardt polyhedron and has vertices of the polyhedron as its four vertices. For, among any four vertices of the Schönhardt polyhedron, at least one pair of vertices from these four vertices must be a diagonal of the polyhedron, which lies entirely outside the polyhedron.
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The vertices of a 600-cell centered at the origin of 4-space, with edges of length (where φ = is the golden ratio), can be given as follows: 16 vertices of the form: :(±, ±, ±, ±), and 8 vertices obtained from :(0, 0, 0, ±1) by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of :(±φ, ±1, ±, 0). Note that the first 16 vertices are the vertices of a tesseract, the second eight are the vertices of a 16-cell, and that all 24 vertices together are vertices of a 24-cell. The final 96 vertices are the vertices of a snub 24-cell, which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.
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Its 180 vertices represent the root vectors of the simple Lie group D10. The vertices can be seen in 3 hyperplanes, with the 45 vertices rectified 9-simplices facets on opposite sides, and 90 vertices of an expanded 9-simplex passing through the center. When combined with the 20 vertices of the 9-orthoplex, these vertices represent the 200 root vectors of the simple Lie group B10.
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The hypergraph is recovered by defining the vertices as the variable-vertices and the hyperedges as the sets of variable-vertices connected to each constraint- vertex.
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When the query vertices belong to different communities, the non-query vertices that form the minimum Wiener connector contain vertices adjacent to edges that bridge the different communities. These vertices span a structural hole in the graph and are important.
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Its 144 vertices represent the root vectors of the simple Lie group D9. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex passing through the center. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B9 and C9 simple Lie groups.
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Its 84 vertices represent the root vectors of the simple Lie group D7. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B7 and C7 simple Lie groups.
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Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope,Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. (Page 29) and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices. However, in graph theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. There is also a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, and if a polygon is approximated by a smooth curve, there will be a point of extreme curvature near each polygon vertex.
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Comparison of several common line generalization algorithms. Gray: original line (394 vertices), orange: 1973 Douglas-Peucker simplification (11 vertices), blue: 2002 PAEK smoothing (483 vertices), red: 2004 Zhou-Jones simplification (31 vertices). All were run with the same tolerance parameters. Another early focus of generalization research, simplification is the removal of vertices in lines and area boundaries.
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Further, since each vertex was previously shared with three other cubes, the vertex would split into 12 rather than three new vertices. However, since some of the shrunken faces continues to be shared, certain pairs of these 12 potential vertices are identical to each other, and therefore only 6 new vertices are created from each original vertex (hence the cantellated tesseract's 96 vertices compared to the tesseract's 16). These six new vertices form the vertices of an octahedron—16 octahedra, since the tesseract had 16 vertices.
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The vertices of the first stellation of the rhombic dodecahedron include the 12 vertices of the cuboctahedron, together with eight additional vertices (the degree-3 vertices of the rhombic dodecahedron). Escher's solid has six additional vertices, at the center points of the square faces of the cuboctahedron (the degree-4 vertices of the rhombic dodecahedron). In the first stellation of the rhombic dodecahedron, these six points are not vertices, but are instead the midpoints of pairs of edges that cross at right angles at these points. The first stellation of the rhombic dodecahedron has 12 hexagonal faces, 36 edges, and 20 vertices, yielding an Euler characteristic of 20 − 36 + 12 = −4.
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A n-dimensional simplex has the minimum n+1 vertices, with all pairs of vertices connected by edges, all triples of vertices defining faces, all quadruples of points defining tetrahedral cells, etc.
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For partial 3-trees there are four forbidden minors: the complete graph on five vertices, the octahedral graph with six vertices, the eight-vertex Wagner graph, and the pentagonal prism with ten vertices..
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The folded cube graph of order k (containing 2k − 1 vertices) may be formed by adding edges between opposite pairs of vertices in a hypercube graph of order k − 1\. (In a hypercube with 2n vertices, a pair of vertices are opposite if the shortest path between them has length n.) It can, equivalently, be formed from a hypercube graph (also) of order k, which has twice as many vertices, by identifying together (or contracting) every opposite pair of vertices.
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In graph theory the distance between two vertices is the length of the shortest path between those vertices.
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If the two vertices are additionally connected by a path of length , i.e. by a single edge, the vertices are called adjacent. A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices.
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Up to a pair of vertices in the top tree \Re can be called as External Boundary Vertices, they can be thought of as Boundary Vertices of the cluster which represents the entire top tree.
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Bridges are closely related to the concept of articulation vertices, vertices that belong to every path between some pair of other vertices. The two endpoints of a bridge are articulation vertices unless they have a degree of 1, although it may also be possible for a non-bridge edge to have two articulation vertices as endpoints. Analogously to bridgeless graphs being 2-edge-connected, graphs without articulation vertices are 2-vertex-connected. In a cubic graph, every cut vertex is an endpoint of at least one bridge.
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The complete list of all trees on 2,3,4 labeled vertices: 2^{2-2}=1 tree with 2 vertices, 3^{3-2}=3 trees with 3 vertices and 4^{4-2}=16 trees with 4 vertices. In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n, the number of trees on n labeled vertices is n^{n-2}. The formula equivalently counts the number of spanning trees of a complete graph with labeled vertices .
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Vertices in graphs are analogous to, but not the same as, vertices of polyhedra: the skeleton of a polyhedron forms a graph, the vertices of which are the vertices of the polyhedron, but polyhedron vertices have additional structure (their geometric location) that is not assumed to be present in graph theory. The vertex figure of a vertex in a polyhedron is analogous to the neighborhood of a vertex in a graph.
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The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B3.
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The hitting set problem is equivalent to the set cover problem: An instance of set cover can be viewed as an arbitrary bipartite graph, with sets represented by vertices on the left, elements of the universe represented by vertices on the right, and edges representing the inclusion of elements in sets. The task is then to find a minimum cardinality subset of left-vertices which covers all of the right- vertices. In the hitting set problem, the objective is to cover the left- vertices using a minimum subset of the right vertices. Converting from one problem to the other is therefore achieved by interchanging the two sets of vertices.
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In the other direction, if a bipartite graph with 14 edges has four vertices on each side, then two vertices on each side must have degree four. Removing these four vertices and their 12 incident edges leaves a nonempty set of edges, any of which together with the four removed vertices forms a K3,3 subgraph.
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Haus vom Nikolaus puzzle has two odd vertices, the path must start at one and end at the other. 2\. Annie Pope's with four odd vertices has no solution. 3\. If there are no odd vertices, the path can start anywhere and forms a closed circuit. 4\. Loose ends are considered vertices of degree 1.
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The smallest 4-crossing cubic graph is the Möbius-Kantor graph, with 16 vertices. The smallest 5-crossing cubic graph is the Pappus graph, with 18 vertices. The smallest 6-crossing cubic graph is the Desargues graph, with 20 vertices. None of the four 7-crossing cubic graphs, with 22 vertices, are well known.
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Set covering is equivalent to the hitting set problem. That is seen by observing that an instance of set covering can be viewed as an arbitrary bipartite graph, with sets represented by vertices on the left, the universe represented by vertices on the right, and edges representing the inclusion of elements in sets. The task is then to find a minimum cardinality subset of left-vertices which covers all of the right- vertices. In the Hitting set problem, the objective is to cover the left- vertices using a minimum subset of the right vertices.
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A skew polygon is a polygon whose vertices are not coplanar. Such a polygon must have at least four vertices; there are no skew triangles. A polyhedron that has positive volume has vertices that are not all coplanar.
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Each of these vertices may link only to vertices from previous epochs. The evolving models above are by no means complete. They can be extended in several ways. First of all, the tails in the models are either static, chosen uniformly from the new vertices, or chosen from the existing vertices proportional to their out-degrees.
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Among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices.
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The detour distance between u and v in C5 is 4 The distance between two vertices in a graph is defined as the minimum of lengths of paths connecting those vertices. The detour distance between two vertices, say, u and v is defined as the length of the longest u-v path in the graph. In the case of a tree the detour distance between any two vertices is same as the distance between the two vertices.
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The minimum Wiener connector behaves like betweenness centrality. When the query vertices belong to the same community, the non-query vertices that form the minimum Wiener connector tend to belong to the same community and have high centrality within the community. Such vertices are likely to be influential vertices playing leadership roles in the community. In a social network, these influential vertices might be good users for spreading information or to target in a viral marketing campaign.
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In graphs with no isolated vertices, every maximal nonblocker (one to which no more vertices can be added) is itself a dominating set.
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A graph (other than a complete graph) has connectivity k if k is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with n vertices has connectivity n − 1, as implied by the first definition. An equivalent definition is that a graph with at least two vertices is k-connected if, for every pair of its vertices, it is possible to find k vertex-independent paths connecting these vertices; see Menger's theorem .
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Illuminating the skeleton of a convex polyhedron from a light source close to one of its faces causes its shadows to form a planar Schlegel diagram. An undirected graph is a system of vertices and edges, each edge connecting two of the vertices. From any polyhedron one can form a graph, by letting the vertices of the graph correspond to the vertices of the polyhedron and by connecting any two graph vertices by an edge whenever the corresponding two polyhedron vertices are the endpoints of an edge of the polyhedron. This graph is known as the skeleton of the polyhedron.
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A rectified cube is a cuboctahedron – edges reduced to vertices, and vertices expanded into new faces A birectified cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices. A rectified cubic honeycomb – edges reduced to vertices, and vertices expanded into new cells. In Euclidean geometry, rectification, also known as critical truncation or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.
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Rectangles have mirror symmetry on mid-edges; rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices.
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Each element will list three vertices from the set of indices of the previously defined vertices given in the element. The indices of the three vertices of the triangles are specified using the , and elements. The order of the vertices must be according to the right-hand rule, such that vertices are listed in counter-clockwise order as viewed from the outside. Each triangle is implicitly assigned a number in the order in which it was declared, starting at zero.
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One method of forming the Kleetope of a polytope is to place a new vertex outside , near the centroid of each facet. If all of these new vertices are placed close enough to the corresponding centroids, then the only other vertices visible to them will be the vertices of the facets from which they are defined. In this case, the Kleetope of is the convex hull of the union of the vertices of and the set of new vertices., p. 217.
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Bipartite Heawood graph. Points are represented by vertices of one color and lines by vertices of the other color. As with any incidence structure, the Levi graph of the Fano plane is a bipartite graph, the vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding point and line are incident. This particular graph is a connected cubic graph (regular of degree 3), has girth 6 and each part contains 7 vertices.
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We may thus pick one of these vertices and call it v2. Now infinitely many vertices of G can be reached from v2 with a simple path which does not include the vertex v1. Each such path must start with one of the finitely many vertices adjacent to v2. So an argument similar to the one above shows that there must be one of those adjacent vertices through which infinitely many vertices can be reached; pick one and call it v3.
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The Kiepert hyperbola is the unique conic which passes through the triangle's three vertices, its centroid, and its circumcenter. Of all triangles contained in a given convex polygon, there exists a triangle with maximal area whose vertices are all vertices of the given polygon.
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Frequent Mesh Problems, www.cs.princeton.edu T-vertices after applying a subdivision modifier. The T-vertices result in cracks in the model because subdivision surfaces only work for meshes with correct topology. Some modeling algorithms such as subdivision surfaces will fail when a model contains T-vertices.
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This walk is a path if it does not repeat any edges, arcs, or vertices, except possibly the first and last vertices. A path is closed if its first and last vertices are the same, and a closed path is a cycle if it does not repeat vertices, except the first and the last. A mixed graph is acyclic if it does not contain a cycle.
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For example, the graph of the figure is a threshold graph. It can be constructed by beginning with a single- vertex graph (vertex 1), and then adding black vertices as isolated vertices and red vertices as dominating vertices, in the order in which they are numbered. Threshold graphs were first introduced by . A chapter on threshold graphs appears in , and the book is devoted to them.
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Once this has been done, the remaining graph must have a nonblocker that includes at least half of its vertices; for instance, if one 2-colors any spanning tree of the graph, each color class is a nonblocker and one of the two color classes includes at least half the vertices. Therefore, if the graph with isolated vertices removed still has 2k or more vertices, the problem can be solved immediately. Otherwise, the remaining graph is a kernel with at most 2k vertices. Dehne et al.
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The octahedral recurrence is a dynamical system defined on the vertices of the octahedral tiling of space. Each octahedron has 6 vertices, and these vertices are labelled in such a way that : a_1b_1 + a_2b_2 = a_3b_3 Here a_i and b_i are the labels of antipodal vertices. A common convention is that a_2,b_2,a_3,b_3 always lie in a central horizontal plane and a_1,b_1 are the top and bottom vertices. The octahedral recurrence is closely related to C. L. Dodgson's method of condensation for computing determinants.
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Let G be a connected, locally finite, infinite graph (this means: any two vertices can be connected by a path, the graph has infinitely many vertices, and each vertex is adjacent to only finitely many other vertices). Then G contains a ray: a simple path (a path with no repeated vertices) that starts at one vertex and continues from it through infinitely many vertices. A common special case of this is that every infinite tree contains either a vertex of infinite degree or an infinite simple path.
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A cycle with an even number of vertices is called an even cycle; a cycle with an odd number of vertices is called an odd cycle.
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A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices.
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In these approaches, the Penrose tiling is viewed as a set of points, its vertices, while the tiles are geometrical shapes obtained by connecting vertices with edges.
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Symmetries of a regular tridecagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices and edge. Gyration orders are given in the center.
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The existence of the Grötzsch graph demonstrates that the assumption of planarity is necessary in Grötzsch's theorem that every triangle-free planar graph is 3-colorable. used a modified version of the Grötzsch graph to disprove a conjecture of on the chromatic number of triangle-free graphs with high degree. Häggkvist's modification consists of replacing each of the five degree-four vertices of the Grötzsch graph by a set of three vertices, replacing each of the five degree-three vertices of the Grötzsch graph by a set of two vertices, and replacing the remaining degree-five vertex of the Grötzsch graph by a set of four vertices. Two vertices in this expanded graph are connected by an edge if they correspond to vertices connected by an edge in the Grötzsch graph.
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The nine blue vertices form a maximum independent set for the Generalized Petersen graph GP(12,4). In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. The size of an independent set is the number of vertices it contains.
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In graph theory, a mathematical discipline, coloring refers to an assignment of colours or labels to vertices, edges and faces of a graph. Defective coloring is a variant of proper vertex coloring. In a proper vertex coloring, the vertices are coloured such that no adjacent vertices have the same colour. In defective coloring, on the other hand, vertices are allowed to have neighbours of the same colour to a certain extent.
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In the mathematical discipline of simplicial homology theory, a simplicial map is a map between simplicial complexes with the property that the images of the vertices of a simplex always span a simplex. Note that this implies that vertices have vertices for images. Simplicial maps are thus determined by their effects on vertices. In particular, there are a finite number of simplicial maps between two given finite simplicial complexes.
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The positions of the new vertices in the mesh are computed based on the positions of nearby old vertices. In some refinement schemes, the positions of old vertices might also be altered (possibly based on the positions of new vertices). This process produces a denser mesh than the original one, containing more polygonal faces. This resulting mesh can be passed through the same refinement scheme again and so on.
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Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid. Truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size.
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A common extension is to hypergraphs, where an edge can connect more than two vertices. A hyperedge is not cut if all vertices are in one partition, and cut exactly once otherwise, no matter how many vertices are on each side. This usage is common in electronic design automation.
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Symmetries of a regular hendecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edge. Gyration orders are given in the center.
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Symmetries of a regular heptadecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.
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Symmetries of a regular enneadecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.
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Symmetries of a regular pentagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.
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An important subclass of the dominating sets is the class of connected dominating sets. If S is a connected dominating set, one can form a spanning tree of G in which S forms the set of non-leaf vertices of the tree; conversely, if T is any spanning tree in a graph with more than two vertices, the non-leaf vertices of T form a connected dominating set. Therefore, finding minimum connected dominating sets is equivalent to finding spanning trees with the maximum possible number of leaves. A total dominating set is a set of vertices such that all vertices in the graph (including the vertices in the dominating set themselves) have a neighbor in the dominating set.
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Combinatorially, one can define a polygon as a set of vertices, a set of edges, and an incidence relation (which vertices and edges touch): two adjacent vertices determine an edge, and dually, two adjacent edges determine a vertex. Then the dual polygon is obtained by simply switching the vertices and edges. Thus for the triangle with vertices {A, B, C} and edges {AB, BC, CA}, the dual triangle has vertices {AB, BC, CA}, and edges {B, C, A}, where B connects AB & BC, and so forth. This is not a particularly fruitful avenue, as combinatorially, there is a single family of polygons (given by number of sides); geometric duality of polygons is more varied, as are combinatorial dual polyhedra.
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The set of vertices reachable from starting vertex form a subgraph with a shape resembling the Greek letter rho (): a path of length from to a cycle of vertices..
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Eigenvector centrality, therefore, can be view as a centrality scoring system for not just one but its neighboring vertices as well. Components :Subgroups, or subsets of vertices, in a disconnected network. Disconnected network means in such network, there is at least a pair of vertices that no path connecting between them at all. Vice verse is known as a connected network, where all vertices within are connected by at least one path.
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One possible generalization of a hypergraph is to allow edges to point at other edges. There are two variations of this generalization. In one, the edges consist not only of a set of vertices, but may also contain subsets of vertices, subsets of subsets of vertices and so on ad infinitum. In essence, every edge is just an internal node of a tree or directed acyclic graph, and vertices are the leaf nodes.
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That is, the vertices of one diagonal on this face are positive and the vertices on the other are negative. Observe that in this case, the signs of the face vertices are insufficient to determine the correct way to triangulate the isosurface. Similarly, an interior ambiguity occurs when the signs of the cube vertices are insufficient to determine the correct surface triangulation, i.e., when multiple triangulations are possible for the same cube configuration.
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A 3-outerplanar graph, the graph of a rhombic dodecahedron. There are four vertices on the outside face, eight vertices on the second layer (light yellow), and two vertices on the third layer (darker yellow). Because of the symmetries of the graph, no other embedding has fewer layers. In graph theory, a k-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k concentric layers.
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In each iteration a refinement heuristic is applied. The merging of vertices induces a map between vertices of a graph and vertices of its coarser graph which is used for the back projection. A rebalancing to ensure the size of the partition may be needed since vertices not belonging to the same partition may be merged. The multi-level technique has shown to significantly improve the results, in terms of both quality and running time.
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Several authors proved sufficient conditions for the case d=r-1, i.e., conditions on the smallest degree of sets of r-1 vertices, in r-uniform hypergraphs with n vertices.
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They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed. Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.
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The smallest cubic graphs with crossing numbers 1–8 and 11 are known . The smallest 1-crossing cubic graph is the complete bipartite graph , with 6 vertices. The smallest 2-crossing cubic graph is the Petersen graph, with 10 vertices. The smallest 3-crossing cubic graph is the Heawood graph, with 14 vertices.
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The name flower snark is sometimes used for J5, a flower snark with 20 vertices and 30 edges. It is one of 6 snarks on 20 vertices . The flower snark J5 is hypohamiltonian. J3 is a trivial variation of the Petersen graph formed by replacing one of its vertices by a triangle.
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An undirected graph that is not connected is called disconnected. An undirected graph G is therefore disconnected if there exist two vertices in G such that no path in G has these vertices as endpoints. A graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected.
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These 12 order-5 vertices can be truncated such that all edges are equal length. The original 30 rhombic faces become non-regular hexagons, and the truncated vertices become regular pentagons. The hexagon faces can be equilateral but not regular with D2 symmetry. The angles at the two vertices with vertex configuration 6.6.
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The automorphism group of the Gray graph is a group of order 1296. It acts transitively on the edges the graph but not on its vertices : there are symmetries taking every edge to any other edge, but not taking every vertex to any other vertex. The vertices that correspond to points of the underlying configuration can only be symmetric to other vertices that correspond to points, and the vertices that correspond to lines can only be symmetric to other vertices that correspond to lines. Therefore, the Gray graph is a semi-symmetric graph, the smallest possible cubic semi- symmetric graph.
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The Levi graphs of projective configurations lead to many important symmetric graphs and cages. The visibility graph of a closed polygon connects each pair of vertices by an edge whenever the line segment connecting the vertices lies entirely in the polygon. It is not known how to test efficiently whether an undirected graph can be represented as a visibility graph. A partial cube is a graph for which the vertices can be associated with the vertices of a hypercube, in such a way that distance in the graph equals Hamming distance between the corresponding hypercube vertices.
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An exhaustive search over all possible subsets of vertices to find the one that induces the connector of minimum Wiener index yields an algorithm that finds the optimum solution in 2^{O(n)} time (that is, exponential time) on graphs with n vertices. In the special case that there are exactly two query vertices, the optimum solution is the shortest path joining the two vertices, so the problem can be solved in polynomial time by computing the shortest path. In fact, for any fixed constant number of query vertices, an optimum solution can be found in polynomial time.
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The blocks are attached to each other at shared vertices called cut vertices or articulation points. Specifically, a cut vertex is any vertex whose removal increases the number of connected components.
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The graph made up of the vertices and edges of the d-dimensional cyclohedron is the flip graph of the centrally symmetric triangulations of a convex polygon with 2d+2 vertices.
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The number of vertices in Δd,k is \tbinom d k . The graph formed by the vertices and edges of a hypersimplex Δd,k is the Johnson graph J(d,k)..
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In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by and disproved by , who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by . The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.
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The map P \to Q is slightly problematic, in the sense that the indices of the P-vertices are naturally odd integers whereas the indices of Q-vertices are naturally even integers. A more conventional approach to the labeling would be to label the vertices of P and Q by integers of the same parity. One can arrange this either by adding or subtracting 1 from each of the indices of the Q-vertices. Either choice is equally canonical.
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For deterministic algorithms, originally conjectured that for all nontrivial graph properties on n vertices, deciding whether a graph possesses this property requires Ω(n2) queries. The non-triviality condition is clearly required because there are trivial properties like "is this a graph?" which can be answered with no queries at all. A scorpion graph. One of the three red path vertices is adjacent to all other vertices and the other two red vertices have no other adjacencies.
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A graph meeting the conditions of Ore's theorem, and a Hamiltonian cycle in it. There are two vertices with degree less than n/2 in the center of the drawing, so the conditions for Dirac's theorem are not met. However, these two vertices are adjacent, and all other pairs of vertices have total degree at least seven, the number of vertices. Ore's theorem is a result in graph theory proved in 1960 by Norwegian mathematician Øystein Ore.
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An undirected graph is a mathematical object consisting of a set of vertices and a set of edges that link pairs of vertices. The two vertices associated with each edge are called its endpoints. The graph is finite when its vertices and edges form finite sets, and infinite otherwise. A graph coloring associates each vertex with a color drawn from a set of colors, in such a way that every edge has two different colors at its endpoints.
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The 6-j symbol is associated with the K4 graph on 4 vertices, the 9-j symbol with the utility graph on 6 vertices (K3,3), and the two distinct (non-isomorphic) 12-j symbols with the Q3 and Wagner graphs on 8 vertices. Symmetry relations are generally representative of the automorphism group of these graphs.
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Notice that the last two vertices, 8 and 9 at the top and bottom center of the "box- cylinder", have four connected vertices rather than five. A general system must be able to handle an arbitrary number of vertices connected to any given vertex. For a complete description of VV meshes see Smith (2006).
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Symmetries of a regular enneagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.
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We start with the triangulated subdivision, and choose an independent set of vertices to be removed. After removing the vertices, we retriangulate the subdivision. Because the subdivision is formed by triangles, a greedy algorithm can find an independent set that contains a constant fraction of the vertices. Therefore, the number of removal steps is O(log n).
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To see this, note that is equal to the closed square in with vertices , and while is a closed "hour glass shaped" shaped subset that intersects the -axis at the origin and is the union of two triangles: one whose vertices are the origin along with and the other triangle whose vertices are the origin along with }.
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The hemicube should not be confused with the demicube – the hemicube is a projective polyhedron, while the demicube is an ordinary polyhedron (in Euclidean space). While they both have half the vertices of a cube, the hemicube is a quotient of the cube, while the vertices of the demicube are a subset of the vertices of the cube.
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Three ideal triangles in the Poincaré disk model Two ideal triangles in the Poincaré half-plane model In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles. The vertices are sometimes called ideal vertices. All ideal triangles are congruent.
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In particular, a complete graph with vertices, denoted , has no vertex cuts at all, but . A vertex cut for two vertices and is a set of vertices whose removal from the graph disconnects and . The local connectivity is the size of a smallest vertex cut separating and . Local connectivity is symmetric for undirected graphs; that is, .
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The Auslander-Reiten quiver of an Artin algebra has a vertex for each indecomposable module and an arrow between vertices if there is an irreducible morphism between the corresponding modules. It has a map τ = D Tr called the translation from the non-projective vertices to the non-injective vertices, where D is the dual and Tr the transpose.
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Solving the version in which guards must be placed on vertices and only vertices need to be guarded is equivalent to solving the dominating set problem on the visibility graph of the polygon.
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A rigid graph is an embedding of a graph in a Euclidean space which is structurally rigid. That is, a graph is rigid if the structure formed by replacing the edges by rigid rods and the vertices by flexible hinges is rigid. A graph that is not rigid is called flexible. More formally, a graph embedding is flexible if the vertices can be moved continuously, preserving the distances between adjacent vertices, with the result that the distances between some nonadjacent vertices are altered.
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An (N, M, D, K, e)-disperser is a bipartite graph with N vertices on the left side, each with degree D, and M vertices on the right side, such that every subset of K vertices on the left side is connected to more than (1 − e)M vertices on the right. An extractor is a related type of graph that guarantees an even stronger property; every (N, M, D, K, e)-extractor is also an (N, M, D, K, e)-disperser.
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To do so, attach a new vertex to each of the subset vertices, which is neither an initial nor desired vertex. Label the vertices on the long path 1 through n, and do the same for the element vertices. Now, the solution consists of 'moving aside' each chosen subset vertex token, correctly placing the labeled vertices from the path, and returning the subset vertex tokens to the initial locations. This is an L-reduction with \alpha = 1/5, \beta = 2.
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Different orderings of the vertices of a graph may cause the greedy coloring to use different numbers of colors, ranging from the optimal number of colors to, in some cases, a number of colors that is proportional to the number of vertices in the graph. For instance, a crown graph (a graph formed from two disjoint sets of vertices } and } by connecting to whenever ) can be a particularly bad case for greedy coloring. With the vertex ordering , a greedy coloring will use colors, one color for each pair . However, the optimal number of colors for this graph is two, one color for the vertices and another for the vertices .
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A graph is vertex-transitive if it has symmetries that map any vertex to any other vertex. In the context of graph enumeration and graph isomorphism it is important to distinguish between labeled vertices and unlabeled vertices. A labeled vertex is a vertex that is associated with extra information that enables it to be distinguished from other labeled vertices; two graphs can be considered isomorphic only if the correspondence between their vertices pairs up vertices with equal labels. An unlabeled vertex is one that can be substituted for any other vertex based only on its adjacencies in the graph and not based on any additional information.
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In the mathematical discipline of graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with n vertices can also be defined as the 1-skeleton of an (n-1)-gonal pyramid. Some authors write Wn to denote a wheel graph with n vertices (n ≥ 4); other authors instead use Wn to denote a wheel graph with n+1 vertices (n ≥ 3), which is formed by connecting a single vertex to all vertices of a cycle of length n. In the rest of this article we use the former notation.
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The Rocha–Thatte algorithm is a general algorithm for detecting cycles in a directed graph G by message passing among its vertices, based on the bulk synchronous message passing abstraction. This is a vertex-centric approach in which the vertices of the graph work together for detecting cycles. The bulk synchronous parallel model consists of a sequence of iterations, in each of which a vertex can receive messages sent by other vertices in the previous iteration, and send messages to other vertices. In each pass, each active vertex of G sends a set of sequences of vertices to its out-neighbours as described next.
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Every pseudoforest on a set of n vertices has at most n edges, and every maximal pseudoforest on a set of n vertices has exactly n edges. Conversely, if a graph G has the property that, for every subset S of its vertices, the number of edges in the induced subgraph of S is at most the number of vertices in S, then G is a pseudoforest. 1-trees can be defined as connected graphs with equally many vertices and edges. Moving from individual graphs to graph families, if a family of graphs has the property that every subgraph of a graph in the family is also in the family, and every graph in the family has at most as many edges as vertices, then the family contains only pseudoforests.
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However, the domination number of this graph is actually much higher. It has n2 + 2n + 1 vertices: n2 formed from the product of a leaf in both factors, 2n from the product of a leaf in one factor and the hub in the other factor, and one remaining vertex formed from the product of the two hubs. Each leaf-hub product vertex in G dominates exactly n of the leaf-leaf vertices, so n leaf-hub vertices are needed to dominate all of the leaf-leaf vertices. However, no leaf-hub vertex dominates any other such vertex, so even after n leaf-hub vertices are chosen to be included in the dominating set, there remain n more undominated leaf-hub vertices, which can be dominated by the single hub-hub vertex.
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The extension property can be used to build up isomorphic copies of any finite or countably infinite graph G within the Rado graph, as induced subgraphs. To do so, order the vertices of G, and add vertices in the same order to a partial copy of G within the Rado graph. At each step, the next vertex in G will be adjacent to some set U of vertices in G that are earlier in the ordering of the vertices, and non-adjacent to the remaining set V of earlier vertices in G. By the extension property, the Rado graph will also have a vertex x that is adjacent to all the vertices in the partial copy that correspond to members of U, and non-adjacent to all the vertices in the partial copy that correspond to members of V. Adding x to the partial copy of G produces a larger partial copy, with one more vertex., Proposition 6.
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Dual of gyrobifastigium The dual polyhedron of the gyrobifastigium has 8 faces: 4 isosceles triangles, corresponding to the degree-three vertices of the gyrobifastigium, and 4 parallelograms corresponding to the degree-four equatorial vertices.
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To the extent that the optimal measure depends on the network structure of the most important vertices, a measure which is optimal for such vertices is sub- optimal for the remainder of the network.
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Marston Conder has compiled a Complete list of all connected edge-transitive graphs on up to 47 vertices and a Complete list of all connected edge-transitive bipartite graphs on up to 63 vertices.
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In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset.
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Equivalently, the vertices of every 2-vertex-connected graph G may be arranged into a cyclic order such that adjacent vertices in this order are at distance at most two from each other in G.
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For how vertices are processed on 3D graphics cards, see shader.
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Defining an ‘irregular graph’ was not immediately obvious. In a k-regular graph, all vertices have degree k. In any graph G with more than one vertex, two vertices in G must have the same degree, so an irregular graph cannot be defined as a graph with all vertices of different degrees. One may be tempted then to define an irregular graph as having all vertices of distinct degrees except for two, but these types of graphs are also well understood and thus not interesting.
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The first-order language of graphs is the collection of well-formed sentences in mathematical logic formed from variables representing the vertices of graphs, universal and existential quantifiers, logical connectives, and predicates for equality and adjacency of vertices. For instance, the condition that a graph does not have any isolated vertices may be expressed by the sentence :\forall u:\exists v: u\sim v where the \sim symbol indicates the adjacency relation between two vertices., Section 1.2, "What Is a First Order Theory?", pp. 15–17.
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Start with any vertex v1. Every one of the infinitely many vertices of G can be reached from v1 with a simple path, and each such path must start with one of the finitely many vertices adjacent to v1. There must be one of those adjacent vertices through which infinitely many vertices can be reached without going through v1. If there were not, then the entire graph would be the union of finitely many finite sets, and thus finite, contradicting the assumption that the graph is infinite.
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To reach a configuration where both vertices carry zero particles from a configuration where this is not the case thus necessarily involves steps where at least one of the two vertices is toppled. Consider the last one of these steps. In this step, one of the two vertices has to topple last. Since toppling transfers a grain of sand to every neighboring vertex, this implies that the total number of grains carried by both vertices together cannot be lower than one, which concludes the proof.
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3D model of a great rhombihexacron The great rhombihexacron is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21). It has 24 identical bow-tie-shaped faces, 18 vertices, and 48 edges. It has 12 outer vertices which have the same vertex arrangement as the cuboctahedron, and 6 inner vertices with the vertex arrangement of an octahedron.
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In geometry Mukhopadhyaya's theorem may refer to one of several closely related theorems about the number of vertices of a curve due to . One version, called the Four-vertex theorem, states that a simple convex curve in the plane has at least 4 vertices, and another version states that a simple convex curve in the affine plane has at least 6 affine vertices.
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Example: an octahedron is a birectification of a cube: {3,4} = 2r{4,3}. Another type of truncation, cantellation, cuts edges and vertices, removing the original edges, replacing them with rectangles, removing the original vertices, and replacing them with the faces of the dual of the original regular polyhedra or tiling. Higher dimensional polytopes have higher truncations. Runcination cuts faces, edges, and vertices.
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In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e.
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Two vertices (or edges) of a periodic graph are symmetric if they are in the same orbit of the symmetry group of the graph; in other words, two vertices (or edges) are symmetric if there is a symmetry of the net that moves one onto the other. In chemistry, there is a tendency to refer to orbits of vertices or edges as “kinds” of vertices or edges, with the recognition that from any two vertices or any two edges (similarly oriented) of the same orbit, the geometric graph “looks the same”. Finite colorings of vertices and edges (where symmetries are to preserve colorings) may be employed. The symmetry group of a crystal net will be a (group of restrictions of a) crystallographic space group, and many of the most common crystals are of very high symmetry, i.e.
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These gadgets would then be glued together to form a single graph, a hard instance for the graph problem in consideration.. For instance, the problem of testing 3-colorability of graphs may be proven NP-complete by a reduction from 3-satisfiability of this type. The reduction uses two special graph vertices, labeled as "Ground" and "False", that are not part of any gadget. As shown in the figure, the gadget for a variable x consists of two vertices connected in a triangle with the ground vertex; one of the two vertices of the gadget is labeled with x and the other is labeled with the negation of x. The gadget for a clause consists of six vertices, connected to each other, to the vertices representing the terms t0, t1, and t2, and to the ground and false vertices by the edges shown.
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Let \phi be the golden ratio. The 12 points given by (0, \pm 1, \pm \phi) and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points (\pm 1, \pm 1, \pm 1) together with the points (\pm\phi, \pm 1/\phi, 0) and cyclic permutations of these coordinates. Multiplying all coordinates of the icosacahedron by a factor of (3\phi+12)/19\approx 0.887\,057\,998\,22 gives a slightly smaller icosahedron. The 12 vertices of this icosahedron, together with the vertices of the dodecahedron, are the vertices of a pentakis dodecahedron centered at the origin.
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By symmetry, each edge of the K6 belongs to three perfect matchings. Incidentally, this partitioning of vertices into edge-vertices and matching-vertices shows that the Tutte-Coxeter graph is bipartite. Based on this construction, Coxeter showed that the Tutte–Coxeter graph is a symmetric graph; it has a group of 1440 automorphisms, which may be identified with the automorphisms of the group of permutations on six elements (Coxeter 1958b). The inner automorphisms of this group correspond to permuting the six vertices of the K6 graph; these permutations act on the Tutte–Coxeter graph by permuting the vertices on each side of its bipartition while keeping each of the two sides fixed as a set.
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A cut vertex is a vertex the removal of which would disconnect the remaining graph; a vertex separator is a collection of vertices the removal of which would disconnect the remaining graph into small pieces. A k-vertex-connected graph is a graph in which removing fewer than k vertices always leaves the remaining graph connected. An independent set is a set of vertices no two of which are adjacent, and a vertex cover is a set of vertices that includes at least one endpoint of each edge in the graph. The vertex space of a graph is a vector space having a set of basis vectors corresponding with the graph's vertices.
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The following generalizes Cayley's formula to labelled forests: Let Tn,k be the number of labelled forests on n vertices with k connected components, such that vertices 1, 2, ..., k all belong to different connected components. Then .
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Consider a path with two vertices, P2. Here the only possibility is to label the only edge in the graph 1. The induced labeling on the two vertices are both 1. So P2 is not edge-graceful.
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Elements of polygonal mesh modeling. Objects created with polygon meshes must store different types of elements. These include vertices, edges, faces, polygons and surfaces. In many applications, only vertices, edges and either faces or polygons are stored.
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The network is made of 50 vertices with initial degrees m_0=1.
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The compact form corresponds to the Satake diagram with all vertices blackened.
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All snarks are non-Hamiltonian, and many known snarks are hypohamiltonian: the removal of any single vertex leaves a Hamiltonian subgraph. A hypohamiltonian snark must be bicritical: the removal of any two vertices leaves a 3-edge-colorable subgraph. It has been shown that the number of snarks for a given even number of vertices is bounded below by an exponential function. (Being cubic graphs, all snarks must have an even number of vertices.) OEIS sequence contains the number of non-trivial snarks of 2n vertices for small values of n.
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In the game of Snort, Red and Blue players take turns coloring the vertices of a graph, with the constraint that two vertices that are connected by an edge may not be colored differently. As usual, the last player to make a legal move is the winner. Since a player's moves improve their position by effectively reserving the adjacent vertices for them alone, positions in Snort are typically hot. In contrast, in the closely related game Col, where adjacent vertices may not have the same color, positions are usually cold.
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For 52, 54, 57, 60 and 64 vertices only one example is known. Of these five graphs only the one with 60 vertices is flexible, the other four are rigid.. It is not possible for a regular matchstick graph to have degree greater than four. The smallest 3-regular matchstick graph without triangles (girth ≥ 4) has 20 vertices, as proved by Kurz and Mazzuoccolo.. The smallest known example of a 3-regular matchstick graph of girth 5 has 54 vertices and was first presented by Mike Winkler in 2019..
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The smallest connected half-transitive graph is Holt's graph, with degree 4 and 27 vertices.. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex-transitive and edge-transitive, rather than an arc-transitive graph. Such a definition would include half-transitive graphs, which are excluded under the definition above. A distance-transitive graph is one where instead of considering pairs of adjacent vertices (i.e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart.
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Each of the seven color classes has one edge from the center to a polygon vertex, and three edges perpendicular to it. A complete graph with vertices is edge-colorable with colors when is an even number; this is a special case of Baranyai's theorem. provides the following geometric construction of a coloring in this case: place points at the vertices and center of a regular -sided polygon. For each color class, include one edge from the center to one of the polygon vertices, and all of the perpendicular edges connecting pairs of polygon vertices.
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Mycielskian construction applied to a 5-cycle graph, producing the Grötzsch graph with 11 vertices and 20 edges, the smallest triangle-free 4-chromatic graph . Let the n vertices of the given graph G be v1, v2, . . . , vn. The Mycielski graph μ(G) contains G itself as a subgraph, together with n+1 additional vertices: a vertex ui corresponding to each vertex vi of G, and an extra vertex w. Each vertex ui is connected by an edge to w, so that these vertices form a subgraph in the form of a star K1,n.
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The powers of three give the place values in the ternary numeral system. In graph theory, powers of three appear in the Moon–Moser bound on the number of maximal independent sets of an -vertex graph, and in the time analysis of the Bron–Kerbosch algorithm for finding these sets. Several important strongly regular graphs also have a number of vertices that is a power of three, including the Brouwer–Haemers graph (81 vertices), Berlekamp–van Lint–Seidel graph (243 vertices), and Games graph (729 vertices).For the Brouwer–Haemers and Games graphs, see .
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The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Further, every triangle has a unique Steiner circumellipse, which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area.
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The Barth Sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes). Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly tetrahedral shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular icosidodecahedron. To these 30 icosidodecahedral vertices are added the summit vertices of the 20 tetrahedral shapes. These 20 points themselves are the vertices of a concentric regular dodecahedron circumscribed about the inner icosidodecahedron.
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In four- dimensional geometry the icosidodecahedron appears in the regular 600-cell as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words: the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices.
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The Kneser graph is the complete graph on vertices. The Kneser graph is the complement of the line graph of the complete graph on vertices. The Kneser graph is the odd graph ; in particular is the Petersen graph.
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First iteration of the perpendicular bisector construction An equivalent construction can be obtained by letting the vertices of Q^{(i+1)} be the circumcenters of the 4 triangles formed by selecting combinations of 3 vertices of Q^{(i)} .
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The basic object used in mesh modeling is a vertex, a point in three-dimensional space. Two vertices connected by a straight line become an edge. Three vertices, connected to each other by three edges, define a triangle, which is the simplest polygon in Euclidean space. More complex polygons can be created out of multiple triangles, or as a single object with more than 3 vertices.
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As proved, the maximum independent set of any ideal polyhedron (the largest possible subset of non-adjacent vertices) must have at most half of the vertices of the polyhedron. It can have exactly half only when the vertices can be partitioned into two equal-size independent sets, so that the graph of the polyhedron is a balanced bipartite graph, as it is for an ideal cube.
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In geometry, the Császár polyhedron () is a nonconvex toroidal polyhedron with 14 triangular faces. This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph K_7 onto a topological torus. Of the 35 possible triangles from vertices of the polyhedron, only 14 are faces.
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Simplicial sets can be viewed as a higher-dimensional generalization of directed multigraphs. A simplicial set contains vertices (known as "0-simplices" in this context) and arrows ("1-simplices") between some of these vertices. Two vertices may be connected by several arrows, and directed loops that connect a vertex to itself are also allowed. Unlike directed multigraphs, simplicial sets may also contain higher simplices.
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In graph theory, the metric dimension of a graph G is the minimum cardinality of a subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given value, is NP-complete.
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For example, the tetrahedron is an alternated cube, h{4,3}. Diminishment is a more general term used in reference to Johnson solids for the removal of one or more vertices, edges, or faces of a polytope, without disturbing the other vertices. For example, the tridiminished icosahedron starts with a regular icosahedron with 3 vertices removed. Other partial truncations are symmetry-based; for example, the tetrahedrally diminished dodecahedron.
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If two chords connect opposite vertices of C to vertices at distance four along C, there is again a 4-cycle. The only remaining case is a Möbius ladder formed by connecting each pair of opposite vertices by a chord, which again has a 4-cycle. Since the Petersen graph has girth five, it cannot be formed in this way and has no Hamiltonian cycle.
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Above:A 3:1-coloring of the cycle on 5 vertices, and the corresponding 6:2-coloring. Below: A 5:2 coloring of the same graph. A b-fold coloring of a graph G is an assignment of sets of size b to vertices of a graph such that adjacent vertices receive disjoint sets. An a:b-coloring is a b-fold coloring out of a available colors.
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In geometry, the great rhombihexacron (or great dipteral disdodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21). It has 24 identical bow-tie-shaped faces, 18 vertices, and 48 edges.Great Rhombihexacron—Bulatov Abstract Creations It has 12 outer vertices which have the same vertex arrangement as the cuboctahedron, and 6 inner vertices with the vertex arrangement of an octahedron.
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Compound of five tetrahedra This compound is unusual, in that the dual figure is the enantiomorph of the original. If the faces are twisted to the right, then the vertices are twisted to the left. When we dualise, the faces dualise to right-twisted vertices and the vertices dualise to left-twisted faces, giving the chiral twin. Figures with this property are extremely rare.
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Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph J(n,k) are the k-element subsets of an n-element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains (k-1)-elements.. Both Johnson graphs and the closely related Johnson scheme are named after Selmer M. Johnson.
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Removing any two vertices (yellow) cannot disconnect a three-dimensional polyhedron: one can choose a third vertex (green), and a nontrivial linear function whose zero set (blue) passes through these three vertices, allowing connections from the chosen vertex to the minimum and maximum of the function, and from any other vertex to the minimum or maximum. In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional polyhedra and higher- dimensional polytopes. It states that, if one forms an undirected graph from the vertices and edges of a convex d-dimensional polyhedron or polytope (its skeleton), then the resulting graph is at least d-vertex-connected: the removal of any d − 1 vertices leaves a connected subgraph. For instance, for a three-dimensional polyhedron, even if two of its vertices (together with their incident edges) are removed, for any pair of vertices there will still exist a path of vertices and edges connecting the pair.. Balinski's theorem is named after mathematician Michel Balinski, who published its proof in 1961,.
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The Cartesian coordinates of the 8 vertices of an elongated octahedron, elongated in the x-axis, with edge length 2 are: : ( ±1, 0, ±2 ) : ( ±2, ±1, 0 ). The 2 extra vertices of the deltahedral variation are: : ( 0, ±1, 0 ).
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Many of these results have analogues for balanced bipartite graphs, in which the vertex degrees are compared to the number of vertices on a single side of the bipartition rather than the number of vertices in the whole graph.
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On the other hand, testing whether there is an edge between two given vertices can be determined at once with an adjacency matrix, while requiring time proportional to the minimum degree of the two vertices with the adjacency list..
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In the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids. It has 30 vertices and 60 edges, and is a quartic graph Archimedean graph.
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Its 72 vertices represent the root vectors of the simple Lie group A8.
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For a graph with n vertices, it requires O(\log^2(n)) time.
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Its 56 vertices represent the root vectors of the simple Lie group A7.
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The Voronoi vertices (nodes) are the points equidistant to three (or more) sites.
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There are many relations among the uniform polyhedra.... Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron. Others share the same vertices and edges as other polyhedron. The grouping below exhibit some of these relations.
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In an ellipse with major axis and minor axis , the vertices on the major axis have the smallest radius of curvature of any points, ; and the vertices on the minor axis have the largest radius of curvature of any points, .
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A partition into exactly n-1 complete bipartite graphs is easy to obtain: just order the vertices, and for each vertex except the last, form a star connecting it to all later vertices in the ordering. Other partitions are also possible.
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The degree sum formula implies that every r-regular graph with n vertices has nr/2 edges. In particular, if r is odd then the number of edges must be divisible by r, and the number of vertices must be even.
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Two vertices of the Gosset graph that come from the same copy are adjacent if they correspond to disjoint edges of K8; two vertices that come from different copies are adjacent if they correspond to edges that share a single vertex..
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Up to constant factors, z(n; t) also bounds the number of edges in an n-vertex graph (not required to be bipartite) that has no Kt,t subgraph. For, in one direction, a bipartite graph with z(n; t) edges and with n vertices on each side of its bipartition can be reduced to a graph with n vertices and (in expectation) z(n; t)/4 edges, by choosing n/2 vertices uniformly at random from each side. In the other direction, a graph with n vertices and no Kt,t can be transformed into a bipartite graph with n vertices on each side of its bipartition, twice as many edges, and still no Kt,t by taking its bipartite double cover., Theorem 2.3, p. 310.
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A graph is 5-vertex-connected when there are no five vertices whose removal leaves a disconnected graph. The complete graph is a graph with an edge between every five vertices, and a subdivision of a complete graph modifies this by replacing some of its edges by longer paths. So a graph contains a subdivision of if it is possible to pick out five vertices of , and a set of ten paths connecting these five vertices in pairs without any of the paths sharing vertices or edges with each other. In any drawing of the graph on the Euclidean plane, at least two of the ten paths must cross each other, so a graph G that contains a K5 subdivision cannot be a planar graph.
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Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph G, and the goal is to find a matching containing as many edges as possible, that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible. An important special case of the maximum cardinality matching problem is when G is a bipartite graph, whose vertices V are partitioned between left vertices in X and right vertices in Y, and edges in E always connect a left vertex to a right vertex.
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Courcelle's theorem may also be used with a stronger variation of monadic second-order logic known as MSO2. In this formulation, a graph is represented by a set V of vertices, a set E of edges, and an incidence relation between vertices and edges. This variation allows quantification over sets of vertices or edges, but not over more complex relations on tuples of vertices or edges. For instance, the property of having a Hamiltonian cycle may be expressed in MSO2 by describing the cycle as a set of edges that includes exactly two edges incident to each vertex, such that every nonempty proper subset of vertices has an edge in the putative cycle with exactly one endpoint in the subset.
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A 2-simplex, for instance, can be thought of as a two-dimensional "triangular" shape bounded by a list of three vertices A, B, C and three arrows B -> C, A -> C and A -> B. In general, an n-simplex is an object made up from a list of n + 1 vertices (which are 0-simplices) and n + 1 faces (which are (n − 1)-simplices). The vertices of the i-th face are the vertices of the n-simplex minus the i-th vertex. The vertices of a simplex need not be distinct and a simplex is not determined by its vertices and faces: two different simplices may share the same list of faces (and therefore the same list of vertices), just like two different arrows in a multigraph may connect the same two vertices. Simplicial sets should not be confused with abstract simplicial complexes, which generalize simple undirected graphs rather than directed multigraphs. Formally, a simplicial set X is a collection of sets Xn, n = 0, 1, 2, ..., together with certain maps between these sets: the face maps dn,i : Xn -> Xn−1 (n = 1, 2, 3, ... and 0 ≤ i ≤ n) and degeneracy maps sn,i : Xn->Xn+1 (n = 0, 1, 2, ... and 0 ≤ i ≤ n).
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4-connectivity 8-connectivity A graph, containing vertices and connecting edges, is constructed from relevant input data. The vertices contain information required by the comparison heuristic, while the edges indicate connected 'neighbors'. An algorithm traverses the graph, labeling the vertices based on the connectivity and relative values of their neighbors. Connectivity is determined by the medium; image graphs, for example, can be 4-connected neighborhood or 8-connected neighborhood.
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Animation of a Reuleaux tetrahedron, showing also the tetrahedron from which it is formed. Four balls intersect to form a Reuleaux tetrahedron. The Reuleaux tetrahedron is the intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s. The spherical surface of the ball centered on each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron.
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A graph is trivially a quotient graph of itself (each block of the partition is a single vertex), and the graph consisting of a single point is the quotient graph of any non-empty graph (the partition consisting of a single block of all vertices). The simplest non-trivial quotient graph is one obtained by identifying two vertices (vertex identification); if the vertices are connected, this is called edge contraction.
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In graph-theoretic mathematics, a biregular graph. or semiregular bipartite graph. is a bipartite graph G=(U,V,E) for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in U is x and the degree of the vertices in V is y, then the graph is said to be (x,y)-biregular.
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Unlike the traditional connectivity, the algebraic connectivity is dependent on the number of vertices, as well as the way in which vertices are connected. In random graphs, the algebraic connectivity decreases with the number of vertices, and increases with the average degree.Synchronization and Connectivity of Discrete Complex Systems, Michael Holroyd, International Conference on Complex Systems, 2006. The exact definition of the algebraic connectivity depends on the type of Laplacian used.
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Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions. Topologically it is considered to contain seven vertices. The three vertices considered at infinity (the real projective plane at infinity) correspond directionally to the three vertices of the hemi- octahedron, an abstract polyhedron.
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A comparison between Gouraud shading (linear interpolation) and non-Gouraud shading (perspective correct interpolation) It is a common misconception that Gouraud shading is any interpolation of colors between vertices. For example, perspective correct interpolation. The original paper makes it clear Gouraud shading is specifically linear interpolation of color between vertices. By default most modern GPUs use perspective correct interpolation between vertices which produces a different result than Gouraud shading.
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The Petersen family. The generalized Petersen graph G(n,k) is formed by connecting the vertices of a regular n-gon to the corresponding vertices of a star polygon with Schläfli symbol {n/k}.; . For instance, in this notation, the Petersen graph is G(5,2): it can be formed by connecting corresponding vertices of a pentagon and five-point star, and the edges in the star connect every second vertex.
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A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph. In graph theory, an odd cycle transversal of an undirected graph is a set of vertices of the graph that has a nonempty intersection with every odd cycle in the graph. Removing the vertices of an odd cycle transversal from a graph leaves a bipartite graph as the remaining induced subgraph.
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However, it does not have an equitable (2n + 1)-coloring: any equitable partition of the vertices into that many color classes would have to have exactly two vertices per class, but the two sides of the bipartition cannot each be partitioned into pairs because they have an odd number of vertices. Therefore, the equitable chromatic threshold of this graph is 2n + 2, significantly greater than its equitable chromatic number of two.
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Any convex polyhedron's surface has Euler characteristic :V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces. For example, a cube has 8 vertices and 6 faces, and hence 12 edges.
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The following theorems can be regarded as directed versions: :Ghouila-Houiri (1960). A strongly connected simple directed graph with n vertices is Hamiltonian if every vertex has a full degree greater than or equal to n. :Meyniel (1973). A strongly connected simple directed graph with n vertices is Hamiltonian if the sum of full degrees of every pair of distinct non-adjacent vertices is greater than or equal to .
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In graph theory, a clique cover or partition into cliques of a given undirected graph is a partition of the vertices of the graph into cliques, subsets of vertices within which every two vertices are adjacent. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum k for which a clique cover exists is called the clique cover number of the given graph.
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In this case the vertices of A1 are the free apices of isosceles triangles with apex angles π/2 erected over the sides of the quadrilateral A0. The vertices of the quadrilateral A2 are the midpoints of the sides of the quadrilateral A1. By the PDN theorem, A2 is a square. The vertices of the quadrilateral A1 are the centers of squares erected over the sides of the quadrilateral A0.
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It maintains a list of vertices that have no incoming edges from other vertices that have not already been included in the partially constructed topological ordering; initially this list consists of the vertices with no incoming edges at all. Then, it repeatedly adds one vertex from this list to the end of the partially constructed topological ordering, and checks whether its neighbors should be added to the list. The algorithm terminates when all vertices have been processed in this way. Alternatively, a topological ordering may be constructed by reversing a postorder numbering of a depth-first search graph traversal.
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It may also be constructed from the vertices of a 5-dimensional hypercube, by connecting pairs of vertices whose Hamming distance is exactly two. This construction is an instance of the construction of Frankl–Rödl graphs. It produces two subsets of 16 vertices that are disconnected from each other; both of these half- squares of the hypercube are isomorphic to the 10-regular Clebsch graph. Two copies of the 5-regular Clebsch graph can be produced in the same way from a 5-dimensional hypercube, by connecting pairs of vertices whose Hamming distance is exactly four.
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In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The degree matrix indicates the degree of vertices. The Laplacian matrix is a modified form of the adjacency matrix that incorporates information about the degrees of the vertices, and is useful in some calculations such as Kirchhoff's theorem on the number of spanning trees of a graph. The distance matrix, like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a shortest path between two vertices.
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Two weights that have been used previously with varying success are the number of node-independent paths between each pair of vertices and the total number of paths between vertices weighted by the length of the path. One disadvantage of these weights, however, is that both weighting schemes tend to separate single peripheral vertices from their rightful communities because of the small number of paths going to these vertices. For this reason, their use in hierarchical clustering techniques is far from optimal. Edge betweenness centrality has been used successfully as a weight in the Girvan–Newman algorithm.
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The closed neighborhood of a vertex in a given graph is the set of vertices consisting of itself and all other vertices adjacent to . The vertex is said to be dominated by another vertex when . That is, and are adjacent, and every other neighbor of is also a neighbor of .. call a vertex that is dominated by another vertex an irreducible vertex. A dismantling order or domination elimination ordering of a given graph is an ordering of the vertices such that, if the vertices are removed one-by-one in this order, each vertex (except the last) is dominated.
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An n-sun cannot be strongly chordal, because the cycle u1w1u2w2... has no odd chord. Strongly chordal graphs may also be characterized as the graphs having a strong perfect elimination ordering, an ordering of the vertices such that the neighbors of any vertex that come later in the ordering form a clique and such that, for each i < j < k < l, if the ith vertex in the ordering is adjacent to the kth and the lth vertices, and the jth and kth vertices are adjacent, then the jth and lth vertices must also be adjacent.; , Theorem 5.5.1, p. 77.
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The smallest cubic semi-symmetric graph (that is, one in which each vertex is incident to exactly three edges) is the Gray graph on 54 vertices. It was first observed to be semi-symmetric by . It was proven to be the smallest cubic semi-symmetric graph by Dragan Marušič and Aleksander Malnič.. All the cubic semi-symmetric graphs on up to 768 vertices are known. According to Conder, Malnič, Marušič and Potočnik, the four smallest possible cubic semi-symmetric graphs after the Gray graph are the Iofinova-Ivanov graph on 110 vertices, the Ljubljana graph on 112 vertices,.
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In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed of only one type of polygon) and excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices. "Identical vertices" means that each two vertices are symmetric to each other: A global isometry of the entire solid takes one vertex to the other while laying the solid directly on its initial position.
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Define a circulant numbering of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to in such a way that, if some two vertices numbered and are adjacent, then every two vertices numbered and are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix. Let be an integer that is relatively prime to , and let be any integer. Then the linear function that takes a number to transforms a circulant numbering to another circulant numbering.
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A graph with edges colored to illustrate path H-A-B (green), closed path or walk with a repeated vertex B-D-E-F-D-C-B (blue) and a cycle with no repeated edge or vertex H-D-G-H (red). In graph theory, a cycle in a graph is a non- empty trail in which the only repeated vertices are the first and last vertices. A directed cycle in a directed graph is a non-empty directed trail in which the only repeated vertices are the first and last vertices. A graph without cycles is called an acyclic graph.
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The locations of the vertices of a regular icosahedron can be described using spherical coordinates, for instance as latitude and longitude. If two vertices are taken to be at the north and south poles (latitude ±90°), then the other ten vertices are at latitude ±arctan() ≈ ±26.57°. These ten vertices are at evenly spaced longitudes (36° apart), alternating between north and south latitudes. This scheme takes advantage of the fact that the regular icosahedron is a pentagonal gyroelongated bipyramid, with D5d dihedral symmetry—that is, it is formed of two congruent pentagonal pyramids joined by a pentagonal antiprism.
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The vertices of a snub 24-cell centered at the origin of 4-space, with edges of length 2, are obtained by taking even permutations of :(0, ±1, ±φ, ±φ2) (where φ = (1+)/2 is the golden ratio). These 96 vertices can be found by partitioning each of the 96 edges of a 24-cell into the golden ratio in a consistent manner, in much the same way that the 12 vertices of an icosahedron or "snub octahedron" can be produced by partitioning the 12 edges of an octahedron in the golden ratio. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector.Coxeter, Regular polytopes, 1973 The 96 vertices of the snub 24-cell, together with the 24 vertices of a 24-cell, form the 120 vertices of the 600-cell.
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The independence domination number iγ(G) of a graph G is the maximum, over all independent sets A of G, of the smallest set dominating A. Dominating subsets of vertices requires potentially less vertices than dominating all vertices, so iγ(G) ≤ γ(G) for all graphs G. The inequality can be strict - there are graphs G for which iγ(G) < γ(G). For example, for some integer n, let G be a graph in which the vertices are the rows and columns of an n-by-n board, and two such vertices are connected if and only if they intersect. The only independent sets are sets of only rows or sets of only columns, and each of them can be dominated by a single vertex (a column or a row), so iγ(G)=1. However, to dominate all vertices we need at least one row and one column, so γ(G)=2.
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Vertex cleaving which is the same as vertex splitting, means one vertex is being split into two, where these two new vertices are adjacent to the vertices that the original vertex was adjacent to. This is the reverse operation of vertex identification.
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The truncated 5-cell may be constructed from the 5-cell by truncating its vertices at 1/3 of its edge length. This transforms the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned near the original vertices.
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Fig.2 Non-Orthogonal Grids Fig.3 Orthogonal Grids a) Structured curvilinear grid arrangements (vertices having similar neighborhood). b) Unstructured grid arrangements (vertices having variation in neighborhood). Structured Curvilinear grids 1) Grid points are identified at the intersection of co-ordinate line.
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Geometry classes support modelling points, linestrings, polygons, and collections. Geometries are linear, in the sense that boundaries are implicitly defined by linear interpolation between vertices. Geometries are embedded in the 2-dimensional Euclidean plane. Geometry vertices may also carry a Z value.
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A complete graph with five vertices and ten edges. Each vertex has an edge to every other vertex. A complete graph is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges.
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Given a partial geometry P, where two points determine at most one line, a collinearity graph of P is a graph whose vertices are the points of P, where two vertices are adjacent if and only if they determine a line in P.
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However, it is closely related to a different binary search tree on the same set of vertices, the Stern–Brocot tree: the vertices at each level of the two trees coincide, and are related to each other by a bit-reversal permutation.
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The midpoint of a segment connecting a hyperbola's vertices is the center of the hyperbola.
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The split form corresponds to the Satake diagram with no vertices blackened and no arrows.
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The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).
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The medial graph of the Herschel graph is a 4-regular planar graph with no Hamiltonian decomposition. The shaded regions correspond to the vertices of the underlying Herschel graph. The Herschel graph also provides an example of a polyhedral graph for which the medial graph cannot be decomposed into two edge-disjoint Hamiltonian cycles. The medial graph of the Herschel graph is a 4-regular graph with 18 vertices, one for each edge of the Herschel graph; two vertices are adjacent in the medial graph whenever the corresponding edges of the Herschel graph are consecutive on one of its faces.. It is 4-vertex-connected and essentially 6-edge-connected, meaning that for every partition of the vertices into two subsets of at least two vertices, there are at least six edges crossing the partition.
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Different choices of the sequence of vertices will typically produce different colorings of the given graph, so much of the study of greedy colorings has concerned how to find a good ordering. There always exists an ordering that produces an optimal coloring, but although such orderings can be found for many special classes of graphs, they are hard to find in general. Commonly used strategies for vertex ordering involve placing higher-degree vertices earlier than lower-degree vertices, or choosing vertices with fewer available colors in preference to vertices that are less constrained. Variations of greedy coloring choose the colors in an online manner, without any knowledge of the structure of the uncolored part of the graph, or choose other colors than the first available in order to reduce the total number of colors.
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The Co atom is surrounded by four NH3 and two Cl ligands at the vertices of an octahedron. The green isomer is "trans" with the two Cl ligands at opposite vertices, and the purple is "cis" with the two Cl at adjacent vertices. Werner also prepared complexes with optical isomers, and in 1914 he reported the first synthetic chiral compound lacking carbon, known as hexol with formula [Co(Co(NH3)4(OH)2)3]Br6.
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If a countably infinite graph G has no odd-degree vertices, then it may be written as a union of disjoint (finite) simple cycles if and only if every finite subgraph of G can be extended (by adding more edges and vertices of G) to a finite Eulerian graph. In particular, every countably infinite graph with only one end and with no odd vertices can be written as a union of disjoint cycles .
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Removing vertices from a path graph can split the remaining graph into as many as connected components. The maximum ratio of components to removed vertices is achieved by removing one vertex (from the interior of the path) and splitting it into two components. Therefore, paths are -tough. In contrast, removing vertices from a cycle graph leaves at most remaining connected components, and sometimes leaves exactly connected components, so a cycle is -tough.
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In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope has vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a cylinder. Regular infinite skew polygons exist in the Petrie polygons of the affine and hyperbolic Coxeter groups.
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Both Tietze's graph and the Petersen graph are maximally nonhamiltonian: they have no Hamiltonian cycle, but any two non- adjacent vertices can be connected by a Hamiltonian path. Tietze's graph and the Petersen graph are the only 2-vertex-connected cubic non-Hamiltonian graphs with 12 or fewer vertices. Unlike the Petersen graph, Tietze's graph is not hypohamiltonian: removing one of its three triangle vertices forms a smaller graph that remains non-Hamiltonian.
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Example of Exact Coloring with 7 colors and 14 vertices In graph theory, an exact coloring is a (proper) vertex coloring in which every pair of colors appears on exactly one pair of adjacent vertices. That is, it is a partition of the vertices of the graph into disjoint independent sets such that, for each pair of distinct independent sets in the partition, there is exactly one edge with endpoints in each set...
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The graph with only one vertex and no edges is called the trivial graph. A graph with only vertices and no edges is known as an edgeless graph. The graph with no vertices and no edges is sometimes called the null graph or empty graph, but the terminology is not consistent and not all mathematicians allow this object. Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable.
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The dual of a cube is an octahedron. Vertices of one correspond to faces of the other, and edges correspond to each other. In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other., "Basic notions about stellation and duality", p. 1.
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The surface vertices are the points where each optical surface crosses the optical axis. They are important primarily because they are the physically measurable parameters for the position of the optical elements, and so the positions of the cardinal points must be known with respect to the vertices to describe the physical system. In anatomy, the surface vertices of the eye's lens are called the anterior and posterior poles of the lens.
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Total coloring is a type of coloring on the vertices and edges of a graph. When used without any qualification, a total coloring is always assumed to be proper in the sense that no adjacent vertices, no adjacent edges, and no edge and its end-vertices are assigned the same color. The total chromatic number χ″(G) of a graph G is the fewest colors needed in any total coloring of G.
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Another kernelization algorithm achieving that bound is based on what is known as the crown reduction rule and uses alternating path arguments. give a kernel based on the crown reduction that has 3k vertices. The 2k vertex bound is a bit more involved and folklore. The currently best known kernelization algorithm in terms of the number of vertices is due to and achieves 2k-c\log k vertices for any fixed constant c.
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Each tesseract-inscribed 3-sphere kisses a surrounding shell of 24 3-spheres, 16 at the vertices of the tesseract and 8 inscribed in the adjacent tesseracts. These 24 kissing points are the vertices of a 24-cell of radius (and edge length) 1/2.
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When these two chains meet in the middle, the simultaneous swap causes adjacent yellow and green vertices in this middle area (such as the vertices represented by the upper yellow and green regions in the figure) to both become red, producing an invalid coloring.
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Its vertex figure has C3v symmetry and has 26 triangular faces, 39 edges, and 15 vertices.
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Femisphere The femisphere is a solid that has one single surface, two edges, and four vertices.
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The tetrahedron's center of mass computes as the arithmetic mean of its four vertices, see Centroid.
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We define an undirected graph to be a set of vertices and edges such that each edge has two vertices (which may coincide) as endpoints. That is, we allow multiple edges (edges with the same pair of endpoints) and loops (edges whose two endpoints are the same vertex). A subgraph of a graph is the graph formed by any subsets of its vertices and edges such that each edge in the edge subset has both endpoints in the vertex subset. A connected component of an undirected graph is the subgraph consisting of the vertices and edges that can be reached by following edges from a single given starting vertex.
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A regular polygon has an inscribed circle which is tangent to each side of the polygon at its midpoint. In a regular polygon with an even number of sides, the midpoint of a diagonal between opposite vertices is the polygon's center. The midpoint-stretching polygon of a cyclic polygon (a polygon whose vertices all fall on the same circle) is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the circular arcs between the vertices of .. Iterating the midpoint-stretching operation on an arbitrary initial polygon results in a sequence of polygons whose shapes converge to that of a regular polygon.
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The graph of a convex polytope P is any graph whose vertices are in bijection with the vertices of P in such a way that any two vertices of the graph are joined by an edge if and only if the two corresponding vertices of P are joined by an edge of the polytope. The diameter of P, denoted \delta(P), is the diameter of any one of its graphs. These definitions are well-defined since any two graphs of the same polytope must be isomorphic as graphs. We may then state the Hirsch conjecture as follows: Conjecture Let P be a d-dimensional convex polytope with n facets.
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In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, or that the vertices lie within a single symmetry orbit.
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A solid angle of π sr is one quarter of that subtended by all of space. When all the solid angles at the vertices of a tetrahedron are smaller than π sr, O lies inside the tetrahedron, and because the sum of distances from O to the vertices is a minimum, O coincides with the geometric median, M, of the vertices. In the event that the solid angle at one of the vertices, v, measures exactly π sr, then O and M coincide with v. If however, a tetrahedron has a vertex, v, with solid angle greater than π sr, M still corresponds to v, but O lies outside the tetrahedron.
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Therefore, a well-covered graph is, equivalently, a graph in which every maximal independent set has the same size, or a graph in which every maximal independent set is maximum. In the original paper defining well-covered graphs, these definitions were restricted to connected graphs,. although they are meaningful for disconnected graphs as well. Some later authors have replaced the connectivity requirement with the weaker requirement that a well-covered graph must not have any isolated vertices.. For both connected well-covered graphs and well-covered graphs without isolated vertices, there can be no essential vertices, vertices which belong to every minimum vertex cover.
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In the similarity graph, the more edges exist for a given number of vertices, the more similar such a set of vertices are between each other. In other words, if we try to disconnect a similarity graph by removing edges, the more edges we need to remove before the graph becomes disconnected, the more similar the vertices in this graph. Minimum cut is a minimum set of edges without which the graph will become disconnected. HCS clustering algorithm finds all the subgraphs with n vertices such that the minimum cut of those subgraphs contain more than n/2 edges, and identifies them as clusters.
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An outerplanar graph (or 1-outerplanar graph) has all of its vertices on the unbounded (outside) face of the graph. A 2-outerplanar graph is a planar graph with the property that, when the vertices on the unbounded face are removed, the remaining vertices all lie on the newly formed unbounded face. And so on. More formally, a graph is k-outerplanar if it has a planar embedding such that, for every vertex, there is an alternating sequence of at most k faces and k vertices of the embedding, starting with the unbounded face and ending with the vertex, in which each consecutive face and vertex are incident to each other.
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A clique cover of a graph G may be seen as a graph coloring of the complement graph of G, the graph on the same vertex set that has edges between non-adjacent vertices of G. Like clique covers, graph colorings are partitions of the set of vertices, but into subsets with no adjacencies (independent sets) rather than cliques. A subset of vertices is a clique in G if and only if it is an independent set in the complement of G, so a partition of the vertices of G is a clique cover of G if and only if it is a coloring of the complement of G.
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It is exactly-2-colorable by the partition X = {1} and Y = {2,3,4}. However, it is not 3-partite: in every partition of V into 3 subsets, at least one subset contains two vertices, and thus at least one hyperedge contains two vertices from this subset.
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A breadth-first search (BFS) is another technique for traversing a finite graph. BFS visits the sibling vertices before visiting the child vertices, and a queue is used in the search process. This algorithm is often used to find the shortest path from one vertex to another.
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It can be seen as a square pyramid without its base. It can be represented symmetrically as a hexagonal or square Schlegel diagram: :160px It has an unexpected property that there are two distinct edges between every pair of vertices – any two vertices define a digon.
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In graph theory, precoloring extension is the problem of extending a graph coloring of a subset of the vertices of a graph, with a given set of colors, to a coloring of the whole graph that does not assign the same color to any two adjacent vertices.
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It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices.. It is labeled V4.8.8 because each isosceles triangle face has two types of vertices: one with 4 triangles, and two with 8 triangles.
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This yields a tetrahedron with edge-length 2, centered at the origin. For the other tetrahedron (which is dual to the first), reverse all the signs. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube.
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Peaks in the spectrum are transformed into vertices in a graph called "spectrum graph". If two vertices have the same mass difference of one or several amino acids, a directed edge will be applied. The SeqMS algorithm, Lutefisk algorithm, Sherenga algorithm are some examples of this type.
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If the faces are equilateral triangles, it is a deltahedron and a Johnson solid (J13). It can be seen as two pentagonal pyramids (J2) connected by their bases. :200px The pentagonal dipyramid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well- covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size.
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If has enough vertices relative to its dimension, then the Kleetope of is dimensionally unambiguous: the graph formed by its edges and vertices is not the graph of a different polyhedron or polytope with a different dimension. More specifically, if the number of vertices of a -dimensional polytope is at least , then is dimensionally unambiguous.; , p. 227. If every -dimensional face of a -dimensional polytope is a simplex, and if , then every -dimensional face of is also a simplex.
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As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage.. The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth as well as odd girth, and again these graphs are cages.
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The median of three vertices in a tree, showing the subtree formed by the union of shortest paths between the vertices. Every tree is a median graph., Proposition 1.26, p. 24. To see this, observe that in a tree, the union of the three shortest paths between pairs of the three vertices a, b, and c is either itself a path, or a subtree formed by three paths meeting at a single central node with degree three.
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The 17280 vertices can be defined as sign and location permutations of: All sign combinations (32): (280×32=8960 vertices) : (4, 2, 2, 2, 2, 0, 0, 0) Half of the sign combinations (128): ((1+8+56)×128=8320 vertices) : (2, 2, 2, 2, 2, 2, 2, 2) : (5, 1, 1, 1, 1, 1, 1, 1) : (3, 3, 3, 1, 1, 1, 1, 1) The edge length is 2 in this coordinate set, and the polytope radius is 4.
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One can construct a bipartite graph in which the vertices on one side are the sets, the vertices on the other side are the elements, and the edges connect a set to the elements it contains. Then, a transversal is equivalent to a perfect matching in this graph. One can construct a hypergraph in which the vertices are the elements, and the hyperedges are the sets. Then, a transversal is equivalent to a vertex cover in a hypergraph.
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A free tree or unrooted tree is a connected undirected graph with no cycles. The vertices with one neighbor are the leaves of the tree, and the remaining vertices are the internal nodes of the tree. The degree of a vertex is its number of neighbors; in a tree with more than one node, the leaves are the vertices of degree one. An unrooted binary tree is a free tree in which all internal nodes have degree exactly three.
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This complementary set induces a matching in G. Each vertex of the independent set is adjacent to n vertices of the matching, and each vertex of the matching is adjacent to n − 1 vertices of the independent set. Because of this decomposition, and because odd graphs are not bipartite, they have chromatic number three: the vertices of the maximum independent set can be assigned a single color, and two more colors suffice to color the complementary matching.
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Any convex polyhedron's surface has Euler characteristic :V - E + F = 2, where is the number of vertices, is the number of edges, and is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, since a cube has 12 edges and 6 faces, the formula implies that it has 8 vertices.
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Definition A d-spindle is a d-dimensional polytope P for which there exist a pair of distinct vertices such that every facet of P contains exactly one of these two vertices. The length of the shortest path between these two vertices is called the length of the spindle. The disproof of the Hirsch conjecture relies on the following theorem, referred to as the strong d-step theorem for spindles. Theorem (Santos) Let P be a d-spindle.
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On a Sudoku board of size n^2\times n^2, the Sudoku graph has n^4 vertices, each with exactly 3n^2-2n-1 neighbors. Therefore, it is a regular graph. For instance, the graph shown in the figure, for a 4\times 4 board, has 16 vertices and is 7-regular. For the most common form of Sudoku, on a 9\times 9 board, the Sudoku graph is a 20-regular graph with 81 vertices.
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The oriented chromatic number of a directed 5-cycle is five. If the cycle is colored by four or fewer colors, then either two adjacent vertices have the same color, or two vertices two steps apart have the same color. In the latter case, the edges connecting these two vertices to the vertex between them are inconsistently oriented: both have the same pair of colors but with opposite orientations. Thus, no coloring with four or fewer colors is possible.
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Graphs with labels attached to edges or vertices are more generally designated as labeled. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called unlabeled. (In the literature, the term labeled may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.) The category of all graphs is the slice category Set ↓ D where D: Set → Set is the functor taking a set s to s × s.
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A graph with three vertices and three edges. A graph (sometimes called undirected graph for distinguishing from a directed graph, or simple graph for distinguishing from a multigraph)See, for instance, Iyanaga and Kawada, 69 J, p. 234 or Biggs, p. 4. is a pair , where is a set whose elements are called vertices (singular: vertex), and is a set of two-sets (sets with two distinct elements) of vertices, whose elements are called edges (sometimes links or lines).
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In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with n vertices is called Cn. The number of vertices in Cn equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.
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In the mathematical field of graph theory, the Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges named after August Ferdinand Möbius and Seligmann Kantor. It can be defined as the generalized Petersen graph G(8,3): that is, it is formed by the vertices of an octagon, connected to the vertices of an eight-point star in which each point of the star is connected to the points three steps away from it.
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The Errera graph, on the other hand, provides a counterexample to Kempe's entire method. When this method is run on the Errera graph, starting with no vertices colored, it can fail to find a valid coloring for the whole graph. Additionally, unlike the Poussin graph, all vertices in the Errera graph have degree five or more. Therefore, on this graph, it is impossible to avoid the problematic cases of Kempe's method by choosing lower-degree vertices.
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This method assumes, in most cases, each edge has the same probability. On the other hand, Bayesian networks are often used for inference and analysis when relationships between each pair of states/events, denoted by vertices, are known. These relationships are usually represented by conditional probabilities among these vertices and are usually obtained from outside of the system. \lambda-connectedness is based on graph theory; however, graph theory only deals with vertices and edges with or without weights.
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Therefore, the set of vertices on the same side of the cut as s automatically forms a closure C. The capacity of the cut equals the weight of all positive-weight vertices minus the weight of the vertices in C, which is minimized when the weight of C is maximized. By the max-flow min-cut theorem, a minimum cut, and the optimal closure derived from it, can be found by solving a maximum flow problem.
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A Hamiltonian path (but not cycle) in the Herschel graph As a bipartite graph that has an odd number of vertices, the Herschel graph does not contain a Hamiltonian cycle (a cycle of edges that passes through each vertex exactly once). For, in any bipartite graph, any cycle must alternate between the vertices on either side of the bipartition, and therefore must contain equal numbers of both types of vertex and must have an even length. Thus, a cycle passing once through each of the eleven vertices cannot exist in the Herschel graph. It is the smallest non-Hamiltonian polyhedral graph, whether the size of the graph is measured in terms of its number of vertices, edges, or faces.. There exist other polyhedral graphs with 11 vertices and no Hamiltonian cycles (notably the Goldner–Harary graph.) but none with fewer edges.
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Graham and Pollak study a more general graph labeling problem, in which the vertices of a graph should be labeled with equal-length strings of the characters "0", "1", and "✶", in such a way that the distance between any two vertices equals the number of string positions where one vertex is labeled with a 0 and the other is labeled with a 1. A labeling like this with no "✶" characters would give an isometric embedding into a hypercube, something that is only possible for graphs that are partial cubes, and in one of their papers Graham and Pollak call a labeling that allows "✶" characters an embedding into a "squashed cube". For each position of the label strings, one can define a complete bipartite graph in which one side of the bipartition consists of the vertices labeled with 0 in that position and the other side consists of the vertices labeled with 1, omitting the vertices labeled "✶". For the complete graph, every two vertices are at distance one from each other, so every edge must belong to exactly one of these complete graphs.
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The main alternative to the adjacency list is the adjacency matrix, a matrix whose rows and columns are indexed by vertices and whose cells contain a Boolean value that indicates whether an edge is present between the vertices corresponding to the row and column of the cell. For a sparse graph (one in which most pairs of vertices are not connected by edges) an adjacency list is significantly more space-efficient than an adjacency matrix (stored as a two-dimensional array): the space usage of the adjacency list is proportional to the number of edges and vertices in the graph, while for an adjacency matrix stored in this way the space is proportional to the square of the number of vertices. However, it is possible to store adjacency matrices more space-efficiently, matching the linear space usage of an adjacency list, by using a hash table indexed by pairs of vertices rather than an array. The other significant difference between adjacency lists and adjacency matrices is in the efficiency of the operations they perform.
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In graph theory, a friendly-index set is a finite set of integers associated with a given undirected graph and generated by a type of graph labeling called a friendly labeling. A friendly labeling of an -vertex undirected graph is defined to be an assignment of the values 0 and 1 to the vertices of with the property that the number of vertices labeled 0 is as close as possible to the number of vertices labeled 1: they should either be equal (for graphs with an even number of vertices) or differ by one (for graphs with an odd number of vertices). Given a friendly labeling of the vertices of , one may also label the edges: a given edge is labeled with a 0 if its endpoints and have equal labels, and it is labeled with a 1 if its endpoints have different labels. The friendly index of the labeling is the absolute value of the difference between the number of edges labeled 0 and the number of edges labeled 1.
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The planar graphs and the apex graphs are linklessly embeddable, as are the graphs obtained by Y-Δ transforms from these graphs. The YΔY reducible graphs are the graphs that can be reduced to a single vertex by Y-Δ transforms, removal of isolated vertices and degree-one vertices, and compression of degree-two vertices; they are also minor-closed, and include all planar graphs. However, there exist linkless graphs that are not YΔY reducible, such as the apex graph formed by connecting an apex vertex to every degree-three vertex of a rhombic dodecahedron.. There also exist linkless graphs that cannot be transformed into an apex graph by Y-Δ transforms, removal of isolated vertices and degree-one vertices, and compression of degree-two vertices: for instance, the ten-vertex crown graph has a linkless embedding, but cannot be transformed into an apex graph in this way. Related to the concept of linkless embedding is the concept of knotless embedding, an embedding of a graph in such a way that none of its simple cycles form a nontrivial knot.
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Chordal bipartite graphs can be recognized in time for a graph with n vertices and m edges.; ; ; .
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Therefore, the solutions to this 2-satisfiability instance correspond one-for-one with the vertices of G.
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Converting from one problem to the other is therefore achieved by interchanging the two sets of vertices.
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Therefore, the diameter of Ramanujan graphs are also bounded logarithmically in terms of the number of vertices.
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Its boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. The edges form 72 flat regular decagons. Each vertex of the 600-cell is a vertex of six such decagons. The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = , 60° = , 72° = , 90° = , 108° = , 120° = , 144° = , and 180° = . Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron, at 60° and 120° the 20 vertices of a dodecahedron, at 72° and 108° again the 12 vertices of an icosahedron, at 90° the 30 vertices of an icosidodecahedron, and finally at 180° the antipodal vertex of V.S.L. van Oss (1899); F. Buekenhout and M. Parker (1998) These can be seen in the H3 Coxeter plane projections with overlapping vertices colored. Just like the icosidodecahedron can be partitioned into 6 central decagons (60 edge = 6×10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). :640px Its vertex figure is an icosahedron, and its dual polytope is the 120-cell, with which it can form a compound.
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The stream processing nature of GPUs remains valid regardless of the APIs used. (See e.g.,) GPUs can only process independent vertices and fragments, but can process many of them in parallel. This is especially effective when the programmer wants to process many vertices or fragments in the same way.
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Hence every Polygon triangulation is a 3-approximation. If the covering is restricted to triangles whose vertices are vertices of the polygon (i.e. Steiner points are not allowed), then the problem is NP-complete. If Steiner points are not allowed and the polygon is in general position (i.e.
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The (red) side edges of tetragonal disphenoid represent a regular zig-zag skew quadrilateral. In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices. The interior surface (or area) of such a polygon is not uniquely defined.
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Other equivalent representations for cellular embeddings include the ribbon graph, a topological space formed by gluing together topological disks for the vertices and edges of an embedded graph, and the graph-encoded map, an edge-colored cubic graph with four vertices for each edge of the embedded graph.
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Sumner's conjecture, named after David Sumner, states that tournaments are universal graphs for polytrees, in the sense that every tournament with 2n − 2 vertices contains every polytree with n vertices as a subgraph. Although it remains unsolved, it has been proven for all sufficiently large values of n.
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These 27 vectors correspond to the vertices of the Schläfli graph; two vertices are adjacent if and only if the corresponding two vectors have 1 as their inner product.. Alternately, this graph can be seen as the complement of the collinearity graph of the generalized quadrangle GQ(2,4).
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In geometry, a diminished rhombic dodecahedron is a rhombic dodecahedron with one or more vertices removed. This article describes diminishing one 4-valence vertex. This diminishment creates one new square face while 4 rhombic faces are reduced to triangles. It has 13 vertices, 24 edges, and 13 faces.
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An involution in the subgroup M22 transposes 8 pairs of co-ordinates. As a permutation matrix in Co0 it has trace 8. It can shown that it moves 80 of the 100 vertices of the Higman-Sims graph. No transposed pair of vertices is an edge in the graph.
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A graph containing a Kempe chain consisting of alternating blue and red vertices Kempe also showed correctly that G can have no vertex of degree 4. As before we remove the vertex v and four-color the remaining vertices. If all four neighbors of v are different colors, say red, green, blue, and yellow in clockwise order, we look for an alternating path of vertices colored red and blue joining the red and blue neighbors. Such a path is called a Kempe chain.
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Then one "flows" the charge by systematically redistributing the charge from a vertex to its neighboring vertices according to a set of rules, the discharging procedure. Since charge is preserved, some vertices still have positive charge. The rules restrict the possibilities for configurations of positively charged vertices, so enumerating all such possible configurations gives an unavoidable set. As long as some member of the unavoidable set is not reducible, the discharging procedure is modified to eliminate it (while introducing other configurations).
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Given such a clique, one can form a covering of space by cubes of side two whose centers have coordinates that, when taken modulo four, are vertices of the clique. The condition that any two vertices of the clique have a coordinate that differs by two implies that cubes corresponding to these vertices do not overlap. The condition that the clique has size 2n implies that the cubes within any period of the tiling have the same total volume as the period itself.
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The Johnson graph is the graph whose vertices are the -element subsets of an -element set, two vertices being adjacent when they meet in a -element set. The Johnson graph is the complement of the Kneser graph . Johnson graphs are closely related to the Johnson scheme, both of which are named after Selmer M. Johnson. The generalized Kneser graph has the same vertex set as the Kneser graph , but connects two vertices whenever they correspond to sets that intersect in or fewer items .
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The Rado graph arises almost surely in the Erdős–Rényi model of a random graph on countably many vertices. Specifically, one may form an infinite graph by choosing, independently and with probability 1/2 for each pair of vertices, whether to connect the two vertices by an edge. With probability 1 the resulting graph is isomorphic to the Rado graph. This construction also works if any fixed probability p not equal to 0 or 1 is used in place of 1/2.
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Because merging two non-adjacent vertices reduces the number of vertices in the resulting graph, the number of operations needed to represent a given graph using the operations defined by Hajós may exceed the number of vertices in . alludes to this when he writes that the sequence of operations is "not always short". , 11.6 Length of Hajós proofs, pp. 184–185, state as an open problem the question of determining the smallest number of steps needed to construct every -vertex graph.
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An L(2,1)-coloring of C6L(2, 1)-coloring is a particular case of L(h, k)-coloring which is in fact a proper coloring. In L(2, 1)-coloring of a graph, G, the vertices of the graph G is colored or labelled in such a way that the adjacent vertices get labels that differ by at least two. Also the vertices that are at a distance of two from each other get labels that differ by at least one.
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Therefore, φ(G) contains medians of all triples of its vertices, and must also be a median graph. In other words, the family of median graphs is closed under the retraction operation., Proposition 1.33, p. 27. A hypercube graph, in which the vertices correspond to all possible k-bit bitvectors and in which two vertices are adjacent when the corresponding bitvectors differ in only a single bit, is a special case of a k-dimensional grid graph and is therefore a median graph.
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This is typically because computer graphics do operations on the vertices at the corners of triangles. With individual triangles, the system has to operate on three vertices for every triangle. In a large mesh, there could be eight or more triangles meeting at a single vertex - by processing those vertices just once, it is possible to do a fraction of the work and achieve an identical effect. In many computer graphics applications it is necessary to manage a mesh of triangles.
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If G is a regular graph of degree d whose edge connectivity is at least d − 1, and G has an even number of vertices, then it has a perfect matching. More strongly, every edge of G belongs to at least one perfect matching. The condition on the number of vertices can be omitted from this result when the degree is odd, because in that case (by the handshaking lemma) the number of vertices is always even., Theorem 4, p. 285.
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Besides the space trade-off, the different data structures also facilitate different operations. Finding all vertices adjacent to a given vertex in an adjacency list is as simple as reading the list. With an adjacency matrix, an entire row must instead be scanned, which takes time. Whether there is an edge between two given vertices can be determined at once with an adjacency matrix, while requiring time proportional to the minimum degree of the two vertices with the adjacency list.
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Construction from the vertices of a truncated octahedron, showing internal rectangles. The coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted. This construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector (ϕ, 1, 0), where ϕ is the golden ratio.
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With three exceptions – the cyclic groups of orders 3, 4, and 5 – every group can be represented as the symmetries of a graph whose vertices have only two orbits. Therefore, the number of vertices in the graph is at most twice the order of the group. With a larger set of exceptions, most finite groups can be represented as the symmetries of a vertex-transitive graph, with a number of vertices equal to the order of the group., Section 4.3.
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In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a set of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common. Fractional graph coloring can be viewed as the linear programming relaxation of traditional graph coloring.
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A path graph or linear graph of order is a graph in which the vertices can be listed in an order v1, v2, …, vn such that the edges are the where i = 1, 2, …, n − 1. Path graphs can be characterized as connected graphs in which the degree of all but two vertices is 2 and the degree of the two remaining vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph.
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Another class of graphs in which the minimum clique cover can be found in polynomial time are the triangle-free graphs. In these graphs, every clique cover consists of a matching (a set of disjoint pairs of adjacent vertices) together with singleton sets for the remaining unmatched vertices. The number of cliques equals the number of vertices minus the number of matched pairs. Therefore, in triangle-free graphs, the minimum clique cover can be found by using an algorithm for maximum matching.
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For rendering, the face list is usually transmitted to the GPU as a set of indices to vertices, and the vertices are sent as position/color/normal structures (in the figure, only position is given). This has the benefit that changes in shape, but not geometry, can be dynamically updated by simply resending the vertex data without updating the face connectivity. Modeling requires easy traversal of all structures. With face-vertex meshes it is easy to find the vertices of a face.
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Few graphs show semi-symmetry: most edge- transitive graphs are also vertex-transitive. The smallest semi-symmetric graph is the Folkman graph, with 20 vertices, which is 4-regular. The three smallest cubic semi-symmetric graphs are the Gray graph, with 54 vertices, this the smallest of the Iofina-Ivanov graphs with 110, and the Ljubljana graph with 112.. It is only for the five Iofina-Ivanov graphs that the symmetry group acts primitively on each partition of the vertices.
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Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l).
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In computing, a distance oracle (DO) is a data structure for calculating distances between vertices in a graph.
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These 12 points project to a hexagram: six vertices around the outer hexagon and six on the inner.
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The claw graph and the path graph on 4 vertices both have the same chromatic polynomial, for example.
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Construct the other two vertices using the compass and the length of the vertex found in step 7a.
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Removing certain triples of vertices from the triakis tetrahedron separates the remaining vertices into multiple connected components. When no such three- vertex separation exists, a polyhedron is said to be 4-connected. Every 4-connected polyhedron has a representation as an ideal polyhedron; for instance this is true of the tetrakis hexahedron, another Catalan solid. Truncating a single vertex from a cube produces a simple polyhedron (one with three edges per vertex) that cannot be realized as an ideal polyhedron: by Miquel's six circles theorem, if seven of the eight vertices of a cube are ideal, the eighth vertex is also ideal, and so the vertices created by truncating it cannot be ideal.
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Matrix structures on the other hand provide faster access for some applications but can consume huge amounts of memory. Implementations of sparse matrix structures that are efficient on modern parallel computer architectures are an object of current investigation. List structures include the edge list, an array of pairs of vertices, and the adjacency list, which separately lists the neighbors of each vertex: Much like the edge list, each vertex has a list of which vertices it is adjacent to. Matrix structures include the incidence matrix, a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the adjacency matrix, in which both the rows and columns are indexed by vertices.
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In a Hanner polytope, every two opposite facets are disjoint, and together include all of the vertices of the polytope, so that the convex hull of the two facets is the whole polytope.. As a simple consequence of this fact, all facets of a Hanner polytope have the same number of vertices as each other (half the number of vertices of the whole polytope). However, the facets may not all be isomorphic to each other. For instance, in the octahedral prism, two of the facets are octahedra, and the other eight facets are triangular prisms. Dually, in every Hanner polytope, every two opposite vertices touch disjoint sets of facets, and together touch all of the facets of the polytope.
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Therefore, the sum of the size of the largest independent set \alpha(G) and the size of a minimum vertex cover \beta(G) is equal to the number of vertices in the graph. A vertex coloring of a graph G corresponds to a partition of its vertex set into independent subsets. Hence the minimal number of colors needed in a vertex coloring, the chromatic number \chi(G), is at least the quotient of the number of vertices in G and the independent number \alpha(G). In a bipartite graph with no isolated vertices, the number of vertices in a maximum independent set equals the number of edges in a minimum edge covering; this is Kőnig's theorem.
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The evolute of a curve will generically have a cusp when the curve has a vertex; other, more degenerate and non-stable singularities may occur at higher-order vertices, at which the osculating circle has contact of higher order than four. Although a single generic curve will not have any higher-order vertices, they will generically occur within a one-parameter family of curves, at the curve in the family for which two ordinary vertices coalesce to form a higher vertex and then annihilate. The symmetry set of a curve has endpoints at the cusps corresponding to the vertices, and the medial axis, a subset of the symmetry set, also has its endpoints in the cusps.
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It is possible to determine whether a graph is strongly chordal in polynomial time, by repeatedly searching for and removing a simple vertex. If this process eliminates all vertices in the graph, the graph must be strongly chordal; otherwise, if this process finds a subgraph without any more simple vertices, the original graph cannot be strongly chordal. For a strongly chordal graph, the order in which the vertices are removed by this process is a strong perfect elimination ordering.. Alternative algorithms are now known that can determine whether a graph is strongly chordal and, if so, construct a strong perfect elimination ordering more efficiently, in time for a graph with n vertices and m edges.; ; .
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In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order v1, v2, …, vn such that the edges are } where i = 1, 2, …, n − 1. Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph. A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more.
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A factor-critical graph, together with perfect matchings of the subgraphs formed by removing one of its vertices. In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph.) is a graph with vertices in which every subgraph of vertices has a perfect matching. (A perfect matching in a graph is a subset of its edges with the property that each of its vertices is the endpoint of exactly one of the edges in the subset.) A matching that covers all but one vertex of a graph is called a near-perfect matching. So equivalently, a factor-critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex.
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That is, the complement is a vertex cover, a set of vertices that includes at least one endpoint of each edge, and is minimal in the sense that none of its vertices can be removed while preserving the property that it is a cover. Minimal vertex covers have been studied in statistical mechanics in connection with the hard-sphere lattice gas model, a mathematical abstraction of fluid-solid state transitions.. Every maximal independent set is a dominating set, a set of vertices such that every vertex in the graph either belongs to the set or is adjacent to the set. A set of vertices is a maximal independent set if and only if it is an independent dominating set.
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The same min-max theorem can be generalized to infinite graphs of finite treewidth, with a definition of treewidth in which the underlying tree is required to be rayless (that is, having no ends). Havens are also closely related to the existence of separators, small sets X of vertices in an n-vertex graph such that every X-flap has at most 2n/3 vertices. If a graph G does not have a k-vertex separator, then every set X of at most k vertices has a (unique) X-flap with more than 2n/3 vertices. In this case, G has a haven of order , in which β(X) is defined to be this unique large X-flap.
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The Buneman graph for five types of mouse. Phylogeny is the inference of evolutionary trees from observed characteristics of species; such a tree must place the species at distinct vertices, and may have additional latent vertices, but the latent vertices are required to have three or more incident edges and must also be labeled with characteristics. A characteristic is binary when it has only two possible values, and a set of species and their characteristics exhibit perfect phylogeny when there exists an evolutionary tree in which the vertices (species and latent vertices) labeled with any particular characteristic value form a contiguous subtree. If a tree with perfect phylogeny is not possible, it is often desired to find one exhibiting maximum parsimony, or equivalently, minimizing the number of times the endpoints of a tree edge have different values for one of the characteristics, summed over all edges and all characteristics.
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When the start state and accept states are ignored, a DFA of n states and an alphabet of size k can be seen as a digraph of n vertices in which all vertices have k out-arcs labeled 1,\ldots, k (a k-out digraph). It is known that when k \ge 2 is a fixed integer, with high probability, the largest strongly connected component (SCC) in such a k-out digraph chosen uniformly at random is of linear size and it can be reached by all vertices. It has also been proven that if k is allowed to increase as n increases, then the whole digraph has a phase transition for strong connectivity similar to Erdős–Rényi model for connectivity. In a random DFA, the maximum number of vertices reachable from one vertex is very close to the number of vertices in the largest SCC with high probability.
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The set of variables involved in a constraint is called the constraint scope. The dual constraint graph is the graph in which the vertices are all constraint scopes involved in the constraints of the problem, and two vertices are connected by an edge if the corresponding scopes have common variables.
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An AMF can represent one object, or multiple objects arranged in a constellation. Each object is described as a set of non-overlapping volumes. Each volume is described by a triangular mesh that references a set of points (vertices). These vertices can be shared among volumes belonging to the same object.
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In graph theory, the Kneser graph (alternatively ) is the graph whose vertices correspond to the -element subsets of a set of elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1955.
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If there are no vertices of odd degree, all Eulerian trails are circuits. If there are exactly two vertices of odd degree, all Eulerian trails start at one of them and end at the other. A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian.
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Alternation or partial truncation removes only some of the original vertices. In partial truncation, or alternation, half of the vertices and connecting edges are completely removed. The operation applies only to polytopes with even-sided faces. Faces are reduced to half as many sides, and square faces degenerate into edges.
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A transitive tournament on 8 vertices. A tournament in which ((a \rightarrow b) and (b \rightarrow c)) \Rightarrow (a \rightarrow c) is called transitive. In other words, in a transitive tournament, the vertices may be (strictly) totally ordered by the edge relation, and the edge relation is the same as reachability.
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Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner.
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Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner.
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The number of vertices is 720° divided by the vertex angle defect. The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular. The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.
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According to the Foster census, the Möbius–Kantor graph is the unique cubic symmetric graph with 16 vertices, and the smallest cubic symmetric graph which is not also distance-transitive.Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.
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There may exist multiple cages for a given combination of r and g. For instance there are three nonisomorphic (3,10)-cages, each with 70 vertices : the Balaban 10-cage, the Harries graph and the Harries–Wong graph. But there is only one (3,11)-cage : the Balaban 11-cage (with 112 vertices).
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The compiler constructs an interference graph, where vertices are variables and an edge connects two vertices if they are needed at the same time. If the graph can be colored with k colors then any set of variables needed at the same time can be stored in at most k registers.
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When the clockwise traversal reaches the starting point, the algorithm returns the sequence of stack vertices as the hull.
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The other four vertices exist at alternate corners of a central cube (a demicube, in this case a tetrahedron).
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Fast algorithms exist to enumerate, up to isomorphism, all regular graphs with a given degree and number of vertices.
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13, 231-237, 1970. The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices..
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At any point during the trip, the Stack Count for the visited leaves and non-terminal vertices cannot subceed.
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Hemicube is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces. Grünbaum also discovered the 11-cell, a four-dimensional self-dual object whose facets are not icosahedra, but are "hemi-icosahedra" -- that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face . The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12.
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A nested triangles graph with 18 vertices In graph theory, a nested triangles graph with n vertices is a planar graph formed from a sequence of n/3 triangles, by connecting pairs of corresponding vertices on consecutive triangles in the sequence. It can also be formed geometrically, by gluing together n/3 − 1 triangular prisms on their triangular faces. This graph, and graphs closely related to it, have been frequently used in graph drawing to prove lower bounds on the area requirements of various styles of drawings.
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To define the half graph on 2n vertices u_1,\dots u_n and v_1,\dots v_n, connect u_i to v_j by an edge whenever i\le j. The same concept can also be defined in the same way for infinite graphs over two copies of any ordered set of vertices. The half graph over the natural numbers (with their usual ordering) has the property that each vertex v_j has finite degree, at most j. The vertices on the other side of the bipartition have infinite degree.
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Here, Pappus observed that a regular dodecahedron and a regular icosahedron could be inscribed in the same sphere such that their vertices all lay on the same 4 circles of latitude, with 3 of the icosahedron's 12 vertices on each circle, and 5 of the dodecahedron's 20 vertices on each circle. This observation has been generalized to higher- dimensional dual polytopes. # An addition by a later writer on another solution to the first problem of the book. Of Book IV the title and preface have been lost.
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Although the classical Gale–Shapley algorithm cannot be implemented as a comparator circuit, Subramanian came up with a different algorithm showing that the problem is in CC. The problem is also CC-complete. Another problem which is CC-complete is lexicographically-first maximal matching. In this problem, we are given a bipartite graph with an order on the vertices, and an edge. The lexicographically-first maximal matching is obtained by successively matching vertices from the first bipartition to the minimal available vertices from the second bipartition.
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In computer graphics, a polygon is a primitive used in modelling and rendering. They are defined in a database, containing arrays of vertices (the coordinates of the geometrical vertices, as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and materials. Any surface is modelled as a tessellation called polygon mesh. If a square mesh has points (vertices) per side, there are n squared squares in the mesh, or 2n squared triangles since there are two triangles in a square.
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Let \phi be the golden ratio. The 12 points given by (0, \pm 1, \pm \phi) and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points (\pm 1, \pm 1, \pm 1) together with the 12 points (0, \pm\phi, \pm 1/\phi) and cyclic permutations of these coordinates. All 32 points together are the vertices of a rhombic triacontahedron centered at the origin.
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The graph of vertices and edges of the 3-3 duoprism has 9 vertices and 18 edges. Like the Berlekamp–van Lint–Seidel graph and the unknown solution to Conway's 99-graph problem, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique square. It is a toroidal graph, a locally linear graph, a strongly regular graph with parameters (9,4,1,2), the 3\times 3 rook's graph, and the Paley graph of order 9.
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The LCF notation is a notation by Joshua Lederberg, Coxeter and Frucht, for the representation of cubic graphs that are Hamiltonian. The two edges along the cycle adjacent to any of the vertices are not written down. Let be the vertices of the graph and describe the Hamiltonian circle along the vertices by the edge sequence . Halting at a vertex , there is one unique vertex at a distance joined by a chord with , : j=i+d_i\quad (\bmod\, p),\quad 2\le d_i\le p-2.
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A clique in a graph is a subset of vertices, all of which are adjacent to each other. A planted clique is a clique created from another graph by adding edges between all pairs of a selected subset of vertices. The planted clique problem can be formalized as a decision problem over a random distribution on graphs, parameterized by two numbers, (the number of vertices), and (the size of the clique). These parameters may be used to generate a graph, by the following random process:.
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The quad-edge structure gets its name from the general mechanism by which they are stored. A single Edge structure conceptually stores references to up to two faces, two vertices, and 4 edges. The four edges stored are the edges starting with the two vertices that are attached to the two stored faces.
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3D model of a pentagonal hexecontahedron In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices.
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Let H be the hypergraph: > { {1,2} , {3,4} , {1,2,3,4} } it is 2-colorable and remains 2-colorable upon removing any number of vertices from it. However, It is not bipartite, since to have exactly one green vertex in each of the first two hyperedges, we must have two green vertices in the last hyperedge.
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If a hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
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John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.The Symmetries of Things, Chapter 20 He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry.
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It has a flat metric outside the set \Sigma images of the vertices. At a point in \Sigma the sum of the angles of the polygons around the vertices which map to it is a positive multiple of 2\pi, and the metric is singular unless the angle is exactly 2\pi.
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The stereographic projection from the sphere to the plane preserves critical points of geodesic curvature. Thus simple closed spherical curves have four vertices. Furthermore, on the sphere vertices of a curve correspond to points where its torsion vanishes. So for space curves a vertex is defined as a point of vanishing torsion.
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In this graph, removing the four red vertices would produce four connected components (depicted in four different colours). However, there is no set of k vertices whose removal leaves more than k components. Therefore, its toughness is exactly 1\. In graph theory, toughness is a measure of the connectivity of a graph.
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Construct a graph with 2n vertices. For each number ai the graph contains two vertices: ui and vi. From each ui, there is only one outgoing edge, which goes to vi and has weight ai. From each vi, there are n outgoing edges, which go to each uj and have weights 0.
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Václav Chvátal proved in 1973 that for all sufficiently large n there exists a hypohamiltonian graph with n vertices. Taking into account subsequent discoveries,; . “sufficiently large” is now known to mean that such graphs exist for all n ≥ 18. A complete list of hypohamiltonian graphs with at most 17 vertices is known:.
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In an undirected graph, an unordered pair of vertices is called connected if a path leads from x to y. Otherwise, the unordered pair is called disconnected. A connected graph is an undirected graph in which every unordered pair of vertices in the graph is connected. Otherwise, it is called a disconnected graph.
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In a hypergraph, an edge can join more than two vertices. An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices. Every graph gives rise to a matroid.
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Klin M.; Lauri J.; Ziv-Av M. "Links between two semisymmetric graphs on 112 vertices through the lens of association schemes", Jour. Symbolic Comput., 47–10, 2012, 1175–1191. In 1972, Bouwer was already talking of a 112-vertices edge- but not vertex-transitive cubic graph found by R. M. Foster, nonetheless unpublished.
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Symmetries of a regular heptagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp.
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Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. By contrast, a hypergraph can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.
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The diameter of the 110-vertex Iofinova-Ivanov graph, the greatest distance between any pair of vertices, is 7. Its radius is likewise 7. Its girth is 10. It is 3-connected and 3-edge-connected: to make it disconnected at least three edges, or at least three vertices, must be removed.
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Instead of upper bounding the number of vertices in a graph in terms of its maximum degree and its diameter, we can calculate via similar methods a lower bound on the number of vertices in terms of its minimum degree and its girth. Suppose G has minimum degree d and girth 2k+1. Choose arbitrarily a starting vertex v, and as before consider the breadth-first search tree rooted at v. This tree must have one vertex at level 0 (v itself), and at least d vertices at level 1. At level 2 (for k > 1), there must be at least d(d-1) vertices, because each vertex at level 1 has at least d-1 remaining adjacencies to fill, and no two vertices at level 1 can be adjacent to each other or to a shared vertex at level 2 because that would create a cycle shorter than the assumed girth.
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An alternative method for improving the basic form of the Bron–Kerbosch algorithm involves forgoing pivoting at the outermost level of recursion, and instead choosing the ordering of the recursive calls carefully in order to minimize the sizes of the sets of candidate vertices within each recursive call. The degeneracy of a graph is the smallest number such that every subgraph of has a vertex with degree or less. Every graph has a degeneracy ordering, an ordering of the vertices such that each vertex has or fewer neighbors that come later in the ordering; a degeneracy ordering may be found in linear time by repeatedly selecting the vertex of minimum degree among the remaining vertices. If the order of the vertices that the Bron–Kerbosch algorithm loops through is a degeneracy ordering, then the set of candidate vertices in each call (the neighbors of that are later in the ordering) will be guaranteed to have size at most .
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Force-directed methods in graph drawing date back to the work of , who showed that polyhedral graphs may be drawn in the plane with all faces convex by fixing the vertices of the outer face of a planar embedding of the graph into convex position, placing a spring-like attractive force on each edge, and letting the system settle into an equilibrium.. Because of the simple nature of the forces in this case, the system cannot get stuck in local minima, but rather converges to a unique global optimum configuration. Because of this work, embeddings of planar graphs with convex faces are sometimes called Tutte embeddings. The combination of attractive forces on adjacent vertices, and repulsive forces on all vertices, was first used by ;. additional pioneering work on this type of force-directed layout was done by .. The idea of using only spring forces between all pairs of vertices, with ideal spring lengths equal to the vertices' graph-theoretic distance, is from ..
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The smallest 3-regular matchstick graph is formed from two copies of the diamond graph placed in such a way that corresponding vertices are at unit distance from each other; its bipartite double cover is the 8-crossed prism graph. In 1986, Heiko Harborth presented the graph that would bear his name, the Harborth Graph. With 104 edges and 52 vertices, is the smallest known example of a 4-regular matchstick graph.. As cited in: It is a rigid graph.. For additional details see Gerbracht's earlier preprint "Minimal Polynomials for the Coordinates of the Harborth Graph" (2006), arXiv:math/0609360. Every 4-regular matchstick graph contains at least 20 vertices.. Examples of 4-regular matchstick graphs are currently known for all number of vertices ≥ 52 except for 53, 55, 56, 58, 59, 61 and 62. The graphs with 54, 57, 65, 67, 73, 74, 77 and 85 vertices were first published in 2016.
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In a hypergraph H = (V, E), each edge of E may contain more than two vertices of V. The degree of a vertex v in V is, as before, the number of edges in E that contain v. But in a hypergraph we can also consider the degree of subsets of vertices: given a subset U of V, deg(U) is the number of edges in E that contain all vertices of U. Thus, the degree of a hypergraph can be defined in different ways depending on the size of subsets whose degree is considered. Formally, for every integer d ≥ 1, degd(H) is the minimum of deg(U) over all subsets U of V that contain exactly d vertices. Thus, deg1(H) corresponds to the definition of a degree of a simple graph, namely the smallest degree of a single vertex; deg2(H) is the smallest degree of a pair of vertices; etc.
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An isolated vertex cannot cover any edges, so in this case v cannot be part of any minimal cover. # If more than k^2 edges remain in the graph, and neither of the previous two rules can be applied, then the graph cannot contain a vertex cover of size k. For, after eliminating all vertices of degree greater than k, each remaining vertex can only cover at most k edges and a set of k vertices could only cover at most k^2 edges. In this case, the instance may be replaced by an instance with two vertices, one edge, and k=0 , which also has no solution. An algorithm that applies these rules repeatedly until no more reductions can be made necessarily terminates with a kernel that has at most k^2 edges and (because each edge has at most two endpoints and there are no isolated vertices) at most 2k^2 vertices.
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Either the vertices will not quite be coplanar, or the faces will have to be distorted slightly away from regularity.
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Second, we must allow a limited number of new vertices to add to each of the embedded graphs with vortices.
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Orthogonal axes in m-dimensional space, vertices of component at n on each axe, the tip at [0,...,0] for .
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The Gale diagram is particularly effective in describing polyhedra whose numbers of vertices are only slightly larger than their dimensions.
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As it does, it stores a convex sequence of vertices on the stack, the ones that have not yet been identified as being within pockets. The points in this sequence are the vertices of a convex polygon (not necessarily the hull of all vertices seen so far) that may have pockets attached to some of its edges. At each step, the algorithm follows a path along the polygon from the stack top to the next vertex that is not in one of the two pockets adjacent to the stack top. Then, while the top two vertices on the stack together with this new vertex are not in convex position, it pops the stack, before finally pushing the new vertex onto the stack.
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In particular, the Kleetope of any three-dimensional polyhedron is a simplicial polyhedron, a polyhedron in which all facets are triangles. Kleetopes may be used to generate polyhedra that do not have any Hamiltonian cycles: any path through one of the vertices added in the Kleetope construction must go into and out of the vertex through its neighbors in the original polyhedron, and if there are more new vertices than original vertices then there are not enough neighbors to go around. In particular, the Goldner–Harary graph, the Kleetope of the triangular bipyramid, has six vertices added in the Kleetope construction and only five in the bipyramid from which it was formed, so it is non-Hamiltonian; it is the simplest possible non-Hamiltonian simplicial polyhedron., p.
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The relation "contains" can be represented by a bipartite graph. The vertices of the graph are divided into two disjoint sets, one representing the subsets in and another representing the elements in X. If a subset contains an element, an edge connects the corresponding vertices in the graph. In the graph representation, an exact cover is a selection of vertices corresponding to subsets such that each vertex corresponding to an element is connected to exactly one selected vertex. For example, the relation "contains" in the detailed example above can be represented by a bipartite graph with 6+7 = 13 vertices: 300px Again, the subcollection = {B, D, F} is an exact cover, since each element is contained in exactly one selected subset, i.e.
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Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs. In particular, every tournament on n vertices contains a transitive subtournament on 1+\lfloor\log_2 n\rfloor vertices.. The proof is simple: choose any one vertex v to be part of this subtournament, and form the rest of the subtournament recursively on either the set of incoming neighbors of v or the set of outgoing neighbors of v, whichever is larger. For instance, every tournament on seven vertices contains a three-vertex transitive subtournament; the Paley tournament on seven vertices shows that this is the most that can be guaranteed . However, showed that this bound is not tight for some larger values of n.
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Consider the complete bipartite graph G = K2,4, having six vertices A, B, W, X, Y, Z such that A and B are each connected to all of W, X, Y, and Z, and no other vertices are connected. As a bipartite graph, G has usual chromatic number 2: one may color A and B in one color and W, X, Y, Z in another and no two adjacent vertices will have the same color. On the other hand, G has list-chromatic number larger than 2, as the following construction shows: assign to A and B the lists {red, blue} and {green, black}. Assign to the other four vertices the lists {red, green}, {red, black}, {blue, green}, and {blue, black}.
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The 26-fullerene graph has D_{3h} prismatic symmetry, the same group of symmetries as the triangular prism. This symmetry group has 12 elements; it has six symmetries that arbitrarily permute the three hexagonal faces of the graph and preserve the orientation of its planar embedding, and another six orientation-reversing symmetries. The number of fullerenes with a given even number of vertices grows quickly in the number of vertices; 26 is the largest number of vertices for which the fullerene structure is unique. The only two smaller fullerenes are the graph of the regular dodecahedron (a fullerene with 20 vertices) and the graph of the truncated hexagonal trapezohedron (a 24-vertex fullerene), which are the two types of cells in the Weaire–Phelan structure.
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For a drawing style in which the vertices are placed on the integer lattice, the area of the drawing may be defined as the area of the smallest axis-aligned bounding box of the drawing: that is, it the product of the largest difference in x-coordinates of two vertices with the largest difference in y-coordinates. For other drawing styles, in which vertices are placed more freely, the drawing may be scaled so that the closest pair of vertices have distance one from each other, after which the area can again be defined as the area of a smallest bounding box of a drawing. Alternatively, the area can be defined as the area of the convex hull of the drawing, again after appropriate scaling..
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As Schnyder observes, the incidence poset of a graph G has order dimension two if and only if the graph is a path or a subgraph of a path. For, in when an incidence poset has order dimension is two, its only possible realizer consists of two total orders that (when restricted to the graph's vertices) are the reverse of each other. Any other two orders would have an intersection that includes an order relation between two vertices, which is not allowed for incidence posets. For these two orders on the vertices, an edge between consecutive vertices can be included in the ordering by placing it immediately following the later of the two edge endpoints, but no other edges can be included.
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In network theory, the Wiener connector is a means of maximizing efficiency in connecting specified "query vertices" in a network. Given a connected, undirected graph and a set of query vertices in a graph, the minimum Wiener connector is an induced subgraph that connects the query vertices and minimizes the sum of shortest path distances among all pairs of vertices in the subgraph. In combinatorial optimization, the minimum Wiener connector problem is the problem of finding the minimum Wiener connector. It can be thought of as a version of the classic Steiner tree problem (one of Karp's 21 NP-complete problems), where instead of minimizing the size of the tree, the objective is to minimize the distances in the subgraph.
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A haven derived from this bramble maps every set X of three or fewer vertices to the unique connected component of G \ X that includes at least one subgraph from the bramble. Havens with the touching definition are closely related to brambles, families of connected subgraphs of a given graph that all touch each other. The order of a bramble is the minimum number of vertices needed in a set of vertices that hits all of the subgraphs in the family. The set of flaps β(X) for a haven of order k (with the touching definition) forms a bramble of order at least k, because any set Y of fewer than k vertices fails to hit the subgraph β(Y).
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Despite the fact that p = 3 would yield a geometrically identical equivalent to the Johnson solid, it lacks a circumscribed sphere that touches all vertices. Its dual, a triangular bipyramid with one of its 4-valence vertices truncated, can be found as cells of the 2-p duoantitegums (duals of the 2-p duoantiprisms).
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Pentagonal stephanoid. This stephanoid has pentagonal dihedral symmetry and has the same vertices as the uniform pentagonal prism. A crown polyhedron or stephanoid is a toroidal polyhedron which is also noble, being both isogonal (equal vertices) and isohedral (equal faces). Crown polyhedra are self-intersecting and topologically self-dual.. See in particular p. 60.
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A regular skew dodecagon seen as zig- zagging edges of a hexagonal antiprism. A skew dodecagon is a skew polygon with 12 vertices and edges but not existing on the same plane. The interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes.
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There are 100 independent sets of size 15 in the Hoffman–Singleton graph. Create a new graph with 100 corresponding vertices, and connect vertices whose corresponding independent sets have exactly 0 or 8 elements in common. The resulting Higman–Sims graph can be partitioned into two copies of the Hoffman-Singleton graph in 352 ways.
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Skew infinite polygons (apeirogons) have vertices which are not all collinear. A zig-zag skew polygon or antiprismatic polygonRegular complex polytopes, p. 6 has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygons in 3 dimensions (and regular skew apeirogons in two dimensions) are always zig-zag.
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In the mathematical field of graph theory, the F26A graph is a symmetric bipartite cubic graph with 26 vertices and 39 edges. It has chromatic number 2, chromatic index 3, diameter 5, radius 5 and girth 6.Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput.
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Certain classes of algebras enjoy both of these properties. The first property is more common, the case of having both is relatively rare. One class that does have both is that of multigraphs. Given two multigraphs and , a homomorphism consists of two functions, one mapping vertices to vertices and the other mapping edges to edges.
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A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.
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The tetrahedron and the Császár polyhedron have no diagonals at all: every pair of vertices in these polyhedra forms an edge. It remains an open question whether there are any other polyhedra (with manifold boundary) without diagonals , although there exist non-manifold surfaces with no diagonals and any number of vertices greater than five .
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See in particular p. 403. It has six vertices, split into two subsets of three vertices, and nine edges, one for each of the nine ways of pairing a vertex from one subset with a vertex from the other subset. The three utilities problem is the question of whether this graph is a planar graph.
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The unitary group SU3(3) (order ) possesses a graph of 36 vertices, in anticipation of the next subgroup. That subgroup is , in which the Hall–Janko group HJ makes its appearance. The aforementioned graph expands to the Hall–Janko graph, with 100 vertices. Next comes , G2(4) being an exceptional group of Lie type.
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Triangle = Tri (three) + Angle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e.
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Figure 2. Vertex- vertex meshes Vertex-vertex meshes represent an object as a set of vertices connected to other vertices. This is the simplest representation, but not widely used since the face and edge information is implicit. Thus, it is necessary to traverse the data in order to generate a list of faces for rendering.
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Balance does not imply bipartiteness. Let H be the hypergraph: > { {1,2} , {3,4} , {1,2,3,4} } it is 2-colorable and remains 2-colorable upon removing any number of vertices from it. However, It is not bipartite, since to have exactly one green vertex in each of the first two hyperedges, we must have two green vertices in the last hyperedge. Bipartiteness does not imply balance. For example, let H be the hypergraph with vertices {1,2,3,4} and edges: > { {1,2,3} , {1,2,4} , {1,3,4} } It is bipartite by the partition X={1}, Y={2,3,4}.
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The performance of the algorithm is strongly determined by the order in which candidate vertices are used to relax other vertices. In fact, if Q is a priority queue, then the algorithm pretty much resembles Dijkstra's. However, since a priority queue is not used here, two techniques are sometimes employed to improve the quality of the queue, which in turn improves the average-case performance (but not the worst-case performance). Both techniques rearranges the order of elements in Q so that vertices closer to the source are processed first.
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Construction of two demicubes (regular tetrahedra, forming a stella octangula) from a single cube. The halved cube graph of order three is the graph of vertices and edges of a single demicube. The halved cube graph of order four includes all of the cube vertices and edges, and all of the edges of the two demicubes. In graph theory, the halved cube graph or half cube graph of order n is the graph of the demihypercube, formed by connecting pairs of vertices at distance exactly two from each other in the hypercube graph.
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Given a graph with a fixed spine ordering for its vertices, drawing these vertices in the same order around a circle and drawing the edges of as line segments produces a collection of chords representing . One can then form a circle graph that has the chords of this diagram as vertices and crossing pairs of chords as edges. A coloring of the circle graph represents a partition of the edges of into subsets that can be drawn without crossing on a single page. Therefore, an optimal coloring is equivalent to an optimal book embedding.
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With index arrays, a mesh is represented by two separate arrays, one array holding the vertices, and another holding sets of three indices into that array which define a triangle. The graphics system processes the vertices first and renders the triangles afterwards, using the index sets working on the transformed data. In OpenGL, this is supported by the glDrawElements() primitive when using Vertex Buffer Object (VBO). With this method, any arbitrary set of triangles sharing any arbitrary number of vertices can be stored, manipulated, and passed to the graphics API, without any intermediary processing.
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In some graphical enumeration problems, the vertices of the graph are considered to be labeled in such a way as to be distinguishable from each other, while in other problems any permutation of the vertices is considered to form the same graph, so the vertices are considered identical or unlabeled. In general, labeled problems tend to be easier.Harary and Palmer, p. 1. As with combinatorial enumeration more generally, the Pólya enumeration theorem is an important tool for reducing unlabeled problems to labeled ones: each unlabeled class is considered as a symmetry class of labeled objects.
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Five is the second Sierpinski number of the first kind, and can be written as S2 = (22) + 1. While polynomial equations of degree and below can be solved with radicals, equations of degree 5 and higher cannot generally be so solved. This is the Abel–Ruffini theorem. This is related to the fact that the symmetric group Sn is a solvable group for and not solvable for . While all graphs with 4 or fewer vertices are planar, there exists a graph with 5 vertices which is not planar: K5, the complete graph with 5 vertices.
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In the ASCII version of the format, the vertices and faces are each described one to a line with the numbers separated by white space. In the binary version, the data is simply packed closely together at the 'endianness' specified in the header and with the data types given in the 'property' records. For the common "property list..." representation for polygons, the first number for that element is the number of vertices that the polygon has and the remaining numbers are the indices of those vertices in the preceding vertex list.
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The vertices of a labeled tree on n vertices are typically given the labels 1, 2, ..., n. A recursive tree is a labeled rooted tree where the vertex labels respect the tree order (i.e., if for two vertices u and v, then the label of u is smaller than the label of v). In a rooted tree, the parent of a vertex v is the vertex connected to v on the path to the root; every vertex has a unique parent except the root which has no parent.
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Each row of Pascal's triangle gives the number of vertices at each distance from a fixed vertex in an n-dimensional cube. For example, in three dimensions, the third row (1 3 3 1) corresponds to the usual three-dimensional cube: fixing a vertex V, there is one vertex at distance 0 from V (that is, V itself), three vertices at distance 1, three vertices at distance and one vertex at distance (the vertex opposite V). The second row corresponds to a square, while larger-numbered rows correspond to hypercubes in each dimension.
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The subgraph isomorphism problem for a fixed subgraph H asks whether H appears as a subgraph of a larger graph G. It may be expressed by a sentence that states the existence of vertices (one for each vertex of H) such that, for each edge of H, the corresponding pair of vertices are adjacent; see picture. As a special case the clique problem (for a fixed clique size) may be expressed by a sentence that states the existence of a number of vertices equal to the clique size all of which are adjacent.
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A 4-coloring of the Petersen graph's edges A 3-coloring of the Petersen graph's vertices The Petersen graph has chromatic number 3, meaning that its vertices can be colored with three colors — but not with two — such that no edge connects vertices of the same color. It has a list colouring with 3 colours, by Brooks' theorem for list colourings. The Petersen graph has chromatic index 4; coloring the edges requires four colors. As a connected bridgeless cubic graph with chromatic index four, the Petersen graph is a snark.
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More formally, a graph G is said to be perfectly orderable if there exists an ordering π of the vertices of G, such that every induced subgraph of G is optimally colored by the greedy algorithm using the subsequence of π induced by the vertices of the subgraph. An ordering π has this property exactly when there do not exist four vertices a, b, c, and d for which abcd is an induced path, a appears before b in the ordering, and c appears after d in the ordering.; .
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Crown graphs with six, eight, and ten vertices. The outer cycle of each graph forms a Hamiltonian cycle; the eight and ten-vertex graphs also have other Hamiltonian cycles. Solutions to the ménage problem may be interpreted in graph-theoretic terms, as directed Hamiltonian cycles in crown graphs. A crown graph is formed by removing a perfect matching from a complete bipartite graph Kn,n; it has 2n vertices of two colors, and each vertex of one color is connected to all but one of the vertices of the other color.
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The Aanderaa–Karp–Rosenberg conjecture concerns implicit graphs given as a set of labeled vertices with a black-box rule for determining whether any two vertices are adjacent. This definition differs from an adjacency labeling scheme in that the rule may be specific to a particular graph rather than being a generic rule that applies to all graphs in a family. Because of this difference, every graph has an implicit representation. For instance, the rule could be to look up the pair of vertices in a separate adjacency matrix.
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Every cycle graph is a circulant graph, as is every crown graph with vertices. The Paley graphs of order (where is a prime number congruent to ) is a graph in which the vertices are the numbers from 0 to and two vertices are adjacent if their difference is a quadratic residue modulo . Since the presence or absence of an edge depends only on the difference modulo of two vertex numbers, any Paley graph is a circulant graph. Every Möbius ladder is a circulant graph, as is every complete graph.
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For keyphrase extraction, it builds a graph using some set of text units as vertices. Edges are based on some measure of semantic or lexical similarity between the text unit vertices. Unlike PageRank, the edges are typically undirected and can be weighted to reflect a degree of similarity. Once the graph is constructed, it is used to form a stochastic matrix, combined with a damping factor (as in the "random surfer model"), and the ranking over vertices is obtained by finding the eigenvector corresponding to eigenvalue 1 (i.e.
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A classic result of graph theory states that a graph of odd order (having an odd number of vertices) always has at least one vertex of even degree. (The statement itself requires zero to be even: the empty graph has an even order, and an isolated vertex has an even degree.) For isolated vertices see p. 149; for groups see p. 311. In order to prove the statement, it is actually easier to prove a stronger result: any odd-order graph has an odd number of even degree vertices.
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Like this algorithm, Tarjan's strongly connected components algorithm also uses depth first search together with a stack to keep track of vertices that have not yet been assigned to a component, and moves these vertices into a new component when it finishes expanding the final vertex of its component. However, in place of the stack P, Tarjan's algorithm uses a vertex-indexed array of preorder numbers, assigned in the order that vertices are first visited in the depth-first search. The preorder array is used to keep track of when to form a new component.
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In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges.. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz's theorem as being exactly the 3-vertex-connected planar graphs.. See in particular Theorem 3, p. 176.
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Several algorithms are based on evaluating this recurrence and the resulting computation tree is sometimes called a Zykov tree. The running time is based on a heuristic for choosing the vertices u and v. The chromatic polynomial satisfies the following recurrence relation :P(G-uv, k)= P(G/uv, k)+ P(G, k) where u and v are adjacent vertices, and G-uv is the graph with the edge removed. P(G - uv, k) represents the number of possible proper colorings of the graph, where the vertices may have the same or different colors.
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In iteration i=3, all the three vertices detect the cycle (2, 3, 4). The algorithm ensures that the cycle is reported only once by emitting the detected cycle only from the vertex with the least identifier value in the ordered sequence, which is the vertex 2 in the example. An example of the execution of the algorithm for detecting cycles by message passing. The total number of iterations of the algorithm is the number of vertices in the longest path in the graph, plus a few more steps for deactivating the final vertices.
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There is a constant-factor approximation algorithm for the minimum Wiener connector problem that runs in time O(q (m \log n + n \log^2 n)) on a graph with n vertices, m edges, and q query vertices, roughly the same time it takes to compute shortest-path distances from the query vertices to every other vertex in the graph. The central approach of this algorithm is to reduce the problem to the vertex-weighted Steiner tree problem, which admits a constant-factor approximation in particular instances related to the minimum Wiener connector problem.
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Singletons adoption: Elements left as singletons by the initial clustering process can be "adopted" by clusters based on similarity to the cluster. If the maximum number of neighbors to a specific cluster is large enough, then it can be added to that cluster. Removing Low Degree Vertices: When the input graph has vertices with low degrees, it is not worthy to run the algorithm since it is computationally expensive and not informative. Alternatively, a refinement of the algorithm can first remove all vertices with a degree lower than certain threshold.
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A vertex cut or separating set of a connected graph is a set of vertices whose removal renders disconnected. The vertex connectivity (where is not a complete graph) is the size of a minimal vertex cut. A graph is called ''-vertex-connected or ''-connected if its vertex connectivity is or greater. More precisely, any graph (complete or not) is said to be -vertex-connected if it contains at least vertices, but does not contain a set of vertices whose removal disconnects the graph; and is defined as the largest such that is -connected.
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In mathematics, the multi-level technique is a technique used to solve the graph partitioning problem. The idea of the multi-level technique is to reduce the magnitude of a graph by merging vertices together, compute a partition on this reduced graph, and finally project this partition on the original graph. In the first phase the magnitude of the graph is reduced by merging vertices. The merging of vertices is done iteratively: of a graph a new coarser graph is created and of this new coarser graph an even more coarse graph is created.
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Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing. Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together.
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For, in a forest, one can always find a constant number of vertices the removal of which leaves a forest that can be partitioned into two smaller subforests with at most 2n/3 vertices each. A linear arrangement formed by recursively partitioning each of these two subforests, placing the separating vertices between them, has logarithmic vertex searching number. The same technique, applied to a tree-decomposition of a graph, shows that, if the treewidth of an n-vertex graph G is t, then the pathwidth of G is O(t log n)., Theorem 6, p.
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For instance, every two-coloring of a five-cycle has a reflection symmetry. In each of these cycles, assigning a unique color to each of two adjacent vertices and using the third color for all remaining vertices results in a three-color distinguishing coloring. However, cycles of six or more vertices have distinguishing colorings with only two colors. That is, Frank Rubin's keyring puzzle requires three colors for rings of three, four or five keys, but only two colors for six or more keys or for two keys.
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Let S be an (a,b)-separator, that is, a vertex subset that separates two nonadjacent vertices a and b. Then S is a minimal (a,b)-separator if no proper subset of S separates a and b. More generally, S is called a minimal separator if it is a minimal separator for some pair (a,b) of nonadjacent vertices. Notice that this is different from minimal separating set which says that no proper subset of S is a minimal (u,v)-separator for any pair of vertices (u,v).
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Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.
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The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.
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The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
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The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
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Escher's solid instead has 48 triangular faces, 72 edges, and 26 vertices, yielding an Euler characteristic of 26 − 72 + 48 = 2.
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The path between the valence-5 vertices is two edges in a row, and then a turn and one more edge.
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The convex hull of the concave polygon's vertices, and that of its edges, contains points that are exterior to the polygon.
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A lobster graph is a tree in which all the vertices are within distance 2 of a central path. Compare caterpillar.
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Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.
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The triangle has as its vertices the two disk centers and one of the two sharp corners of the vesica piscis.
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The conjecture was recently proved, showing that every cubic bridgeless graph with n vertices has at least 2n/3656 perfect matchings..
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However, a smooth curve approximation to a polygon will also have additional vertices, at the points where its curvature is minimal.
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In the mathematical field of graph theory, the Bidiakis cube is a 3-regular graph with 12 vertices and 18 edges.
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In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.
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In geometry, the positive and negative Voronoi poles of a cell in a Voronoi diagram are certain vertices of the diagram.
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The 110-vertex Iofinova-Ivanov graph is, in graph theory, a semi-symmetric cubic graph with 110 vertices and 165 edges.
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Such a choice can only work if no two removed vertices are adjacent, and for each such choice, the subroutine must include in the cover all the vertices outside that are incident to an edge that becomes uncovered by this removal. Using this subroutine in an iterative compression algorithm gives a simple algorithm for vertex cover.
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It is bounded by 16 cells, all of which are regular tetrahedra. It has 32 triangular faces, 24 edges, and 8 vertices. The 24 edges bound 6 squares lying in the 6 coordinate planes. The eight vertices of the 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1).
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The skeleton of the cube (the vertices and edges) form a graph, with 8 vertices, and 12 edges. It is a special case of the hypercube graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid. An extension is the three dimensional k-ary Hamming graph, which for k = 2 is the cube graph.
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He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. These lower symmetries allows degrees of freedom in defining irregular myriagons. Only the g10000 subgroup has no degrees of freedom but can seen as directed edges.
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Let be an instance of the travelling salesman problem. That is, is a complete graph on the set of vertices, and the function assigns a nonnegative real weight to every edge of . According to the triangle inequality, for every three vertices , , and , it should be the case that . Then the algorithm can be described in pseudocode as follows.
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An ellipse has exactly four vertices: two local maxima of curvature where it is crossed by the major axis of the ellipse, and two local minima of curvature where it is crossed by the minor axis. In a circle, every point is both a local maximum and a local minimum of curvature, so there are infinitely many vertices.
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Every vertex of an -dimensional box is connected to edges. If these edges' lengths are , then is the total length of edges incident to the vertex. There are vertices, so we multiply this by ; since each edge, however, meets two vertices, every edge is counted twice. Therefore, we divide by and conclude that there are edges.
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More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. When the end points both lie on a curve (such as a circle), a line segment is called a chord (of that curve).
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Volume is the total number of walks of the given type. The three examples from the previous paragraph fall into this category. Length captures the distance from the given vertex to the remaining vertices in the graph. Freeman's closeness centrality, the total geodesic distance from a given vertex to all other vertices, is the best known example.
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The more subtle limitation is the commonly held fallacy that vertex centrality indicates the relative importance of vertices. Centrality indices are explicitly designed to produce a ranking which allows indication of the most important vertices. This they do well, under the limitation just noted. They are not designed to measure the influence of nodes in general.
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Furthermore, Freeman centralization enables one to compare several networks by comparing their highest centralization scores. This approach, however, is seldom seen in practice. Secondly, the features which (correctly) identify the most important vertices in a given network/application do not necessarily generalize to the remaining vertices. For the majority of other network nodes the rankings may be meaningless.
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In the mathematical discipline of graph theory, the expander walk sampling theorem states that sampling vertices in an expander graph by doing a random walk is almost as good as sampling the vertices independently from a uniform distribution. The earliest version of this theorem is due to , and the more general version is typically attributed to .
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Each color corresponds to a biconnected component. Multi-colored vertices are cut vertices, and thus belong to multiple biconnected components. In graph theory, a biconnected component (sometimes known as a 2-connected component) is a maximal biconnected subgraph. Any connected graph decomposes into a tree of biconnected components called the block-cut tree of the graph.
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He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. These lower symmetries allows degrees of freedoms in defining irregular octacontagons. Only the g80 subgroup has no degrees of freedom but can seen as directed edges.
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Appending an edge and a vertex to P2 gives P3, the path with three vertices. Denote the vertices by v1, v2, and v3. Label the two edges in the following way: the edge (v1, v2) is labeled 1 and (v2, v3) labeled 2. The induced labelings on v1, v2, and v3 are then 1, 0, and 2 respectively.
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In geometry, a vertex (in plural form: vertices or vertexes), often denoted by letters such as P, Q, R, S, is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.
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Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols These properties apply to all regular polygons, whether convex or star. A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points.
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However, the hunt for the smallest planar hypohamiltonian graph continues. This question was first raised by Václav Chvátal in 1973. The answer is provided in 1976 by Carsten Thomassen, who exhibits a 105-vertices construction, the 105-Thomassen graph. In 1979, Hatzel improves this result with a planar hypohamiltonian graph on 57 vertices : the Hatzel graph.
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He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. These lower symmetries allows degrees of freedom in defining irregular hectogons. Only the g100 subgroup has no degrees of freedom but can seen as directed edges.
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Therefore, among the seven vertices of the Moser spindle, there are at most six that have a white copy in M + T, so there must be one of the seven vertices all of whose copies are black. But then the three copies of this vertex form a translate of T.. See also , Problem 40.26, p. 496.
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This kind of graph may be called vertex-labeled. However, for many questions it is better to treat vertices as indistinguishable. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges.) The same remarks apply to edges, so graphs with labeled edges are called edge- labeled.
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A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph., p. 6. In the abstract, all that matters is which pairs of vertices are connected by edges.
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An infinite graph that does not obey the handshaking lemma The handshaking lemma does not apply to infinite graphs, even when they have only a finite number of odd-degree vertices. For instance, an infinite path graph with one endpoint has only a single odd-degree vertex rather than having an even number of such vertices.
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Let K_2 be the complete graph on two vertices (i.e. a single edge). The product graphs K_2 \square K_2, K_2 \times K_2, and K_2 \boxtimes K_2 look exactly like the graph representing the operator. For example, K_2 \square K_2 is a four cycle (a square) and K_2 \boxtimes K_2 is the complete graph on four vertices.
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Alternatively, a Reuleaux triangle may be constructed from an equilateral triangle T by drawing three arcs of circles, each centered at one vertex of T and connecting the other two vertices. Or, equivalently, it may be constructed as the intersection of three disks centered at the vertices of T, with radius equal to the side length of T.
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Various arrangements of capsomeres are: 1) Icosahedral, 2) Helical, and 3) Complex. 1) Icosahedral- An icosahedron is a polyhedron with 12 vertices and 20 faces. Two types of capsomeres constitute the icosahedral capsid: pentagonal (pentons) at the vertices and hexagonal (hexons) at the faces. There are always twelve pentons, but the number of hexons varies among virus groups.
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The Szilassi polyhedron is named after Hungarian mathematician Lajos Szilassi, who discovered it in 1977. The dual to the Szilassi polyhedron, the Császár polyhedron, was discovered earlier by ; it has seven vertices, 21 edges connecting every pair of vertices, and 14 triangular faces. Like the Szilassi polyhedron, the Császár polyhedron has the topology of a torus.
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For three-dimensional simplicial polyhedra the numbers of edges and two-dimensional faces are determined from the number of vertices by Euler's formula, regardless of whether the polyhedron is stacked, but this is not true in higher dimensions. Analogously, the simplicial polytopes that maximize the number of higher-dimensional faces for their number of vertices are the cyclic polytopes.
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A regular polygon with sides has a circumscribed circle upon which all its vertices lie, and the center of the circle is also the center of the polygon. The central angle of the regular polygon is formed at the center by the radii to two adjacent vertices. The measure of this angle is 2\pi/n.
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A polygonal pseudotriangle is a polygon that has exactly three convex vertices. In particular, any triangle, and any nonconvex quadrilateral, is a pseudotriangle. The convex hull of any pseudotriangle is a triangle. The curves along the pseudotriangle boundary between each pair of convex vertices either lie within the triangle or coincide with one of its edges.
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The total graph T(G) of a graph G has as its vertices the elements (vertices or edges) of G, and has an edge between two elements whenever they are either incident or adjacent. The total graph may also be obtained by subdividing each edge of G and then taking the square of the subdivided graph., p. 82.
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The vertex-expansion properties of Johnson graphs, as well as the structure of the corresponding extremal sets of vertices of a given size, are not fully understood. However, an asymptotically tight lower-bound on expansion of large sets of vertices was recently obtained.. In general, determining the chromatic number of a Johnson graph is an open problem.
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The symmetries of a regular pentadecagon as shown with colors on edges and vertices. Lines of reflections are blue. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions. The regular pentadecagon has Dih15 dihedral symmetry, order 30, represented by 15 lines of reflection. Dih15 has 3 dihedral subgroups: Dih5, Dih3, and Dih1.
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The Williot diagram is a graphical method to obtain an approximate value for displacement of a structure which submitted to a certain load. The method consists of, from a graph representation of a structural system, representing the structure's fixed vertices as a single, fixed starting point and from there sequentially adding the neighbouring vertices' relative displacements due to strain.
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The chromatic number of the 110-vertex Iofina-Ivanov graph is 2: its vertices can be 2-colored so that no two vertices of the same color are joined by an edge. Its chromatic index is 3: its edges can be 3-colored so that no two edges of the same color met at a vertex.
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Seth Pettie and Vijaya Ramachandran have found a provably optimal deterministic comparison-based minimum spanning tree algorithm.. The following is a simplified description of the algorithm. # Let r = \log \log \log n, where n is the number of vertices. Find all optimal decision trees on r vertices. This can be done in time O(n) (see Decision trees above).
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Cartesian coordinates for the vertices of a great truncated cuboctahedron centered at the origin are all permutations of : (±1, ±(1−), ±(1−2)).
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The tridyakis icosahedron is the dual polyhedron of the icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.
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Connecting the four 2-degree vertices crosswise creates a cubic graph called a Möbius ladder. Two views of the Möbius ladder M16 .
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An n-gonal form has 3n vertices, 6n edges, and 2+3n faces: 2 regular n-gons, n rhombi, and 2n triangles.
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In all of the algorithms below, m is the number of edges in the graph and n is the number of vertices.
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The order-5 folded cube graph (the 5-regular Clebsch graph) may be constructed by adding edges between opposite pairs of vertices in a 4-dimensional hypercube graph. (In an n-dimensional hypercube, a pair of vertices are opposite if the shortest path between them has n edges.) Alternatively, it can be formed from a 5-dimensional hypercube graph by identifying together (or contracting) every opposite pair of vertices. Another construction, leading to the same graph, is to create a vertex for each element of the finite field GF(16), and connect two vertices by an edge whenever the difference between the corresponding two field elements is a perfect cube. The order-5 halved cube graph (the 10-regular Clebsch graph) is the complement of the 5-regular graph.
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If a cactus is connected, and each of its vertices belongs to at most two blocks, then it is called a Christmas cactus. Every polyhedral graph has a Christmas cactus subgraph that includes all of its vertices, a fact that plays an essential role in a proof by that every polyhedral graph has a greedy embedding in the Euclidean plane, an assignment of coordinates to the vertices for which greedy forwarding succeeds in routing messages between all pairs of vertices.. In topological graph theory, the graphs whose cellular embeddings are all planar are exactly the subfamily of the cactus graphs with the additional property that each vertex belongs to at most one cycle. These graphs have two forbidden minors, the diamond graph and the five-vertex friendship graph.
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A graph with eight vertices, and a tree decomposition of it onto a tree with six nodes. Each graph edge connects two vertices that are listed together at some tree node, and each graph vertex is listed at the nodes of a contiguous subtree of the tree. Each tree node lists at most three vertices, so the width of this decomposition is two. A tree decomposition of a graph G = (V, E) is a tree, T, with nodes X1, ..., Xn, where each Xi is a subset of V, satisfying the following properties section 12.3 (the term node is used to refer to a vertex of T to avoid confusion with vertices of G): # The union of all sets Xi equals V. That is, each graph vertex is contained in at least one tree node.
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The graph of the n-hypercube's edges is isomorphic to the Hasse diagram of the (n−1)-simplex's face lattice. This can be seen by orienting the n-hypercube so that two opposite vertices lie vertically, corresponding to the (n-1)-simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the (n-1)-simplex's facets (n-2 faces), and each vertex connected to those vertices maps to one of the simplex's n-3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices. This relation may be used to generate the face lattice of an (n-1)-simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive.
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The orthocenter of the triangle with vertices in the three vanishing points is the intersection of the optical axis and the image plane.
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Its 42 vertices represent the root vectors of the simple Lie group A6. It is the vertex figure of the 6-simplex honeycomb.
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Records starting with the letter "l" specify the order of the vertices which build a polyline. l v1 v2 v3 v4 v5 v6 ...
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The circles formed in this way are tangent to each other exactly when the vertices they correspond to are connected by an edge.
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For certain sequences of random graphs, Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity.
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But is not balanced. For example, if vertex 1 is removed, we get the restriction of H to {2,3,4}, which has the following hyperedges: > { {2,3} , {2,4} , {3,4} } It is not 2-colorable, since in any 2-coloring there are at least two vertices with the same color, and thus at least one of the hyperedges is monochromatic. Another way to see that H is not balanced is that it contains the odd-length cycle C = (2 - {1,2,3} - 3 - {1,3,4} - 4 - {1,2,4} - 2), and no edge of C contains all three vertices 2,3,4 of C. Tripartiteness does not imply balance. For example, let H be the tripartite hypergraph with vertices {1,2},{3,4},{5,6} and edges: > { {1,3,5}, {2,4,5}, {1,4,6} } It is not balanced since if we remove the vertices 2,3,6, the remainder is: > { {1,5}, {4,5}, {1,4} } which is not colorable since it is a 3-cycle. Another way to see that it is not balanced is that It contains the odd-length cycle C = (1 - {1,3,5} - 5 - {2,4,5} - 4 - {1,4,6} - 1), and no edge of C contains all three vertices 1,4,5 of C.
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This n = 3 case can be given a cubical geometric interpretation (or a graph-theoretic interpretation) as follows: when moving along the edge from \bar x_1 \bar x_2 \bar x_3 to x_1 \bar x_2 \bar x_3, XOR up the functions of the two end-vertices of the edge in order to obtain the coefficient of x_1. To move from \bar x_1 \bar x_2 \bar x_3 to x_1 x_2 \bar x_3 there are two shortest paths: one is a two-edge path passing through x_1 \bar x_2 \bar x_3 and the other one a two-edge path passing through \bar x_1 x_2 \bar x_3. These two paths encompass four vertices of a square, and XORing up the functions of these four vertices yields the coefficient of x_1 x_2. Finally, to move from \bar x_1 \bar x_2 \bar x_3 to x_1 x_2 x_3 there are six shortest paths which are three-edge paths, and these six paths encompass all the vertices of the cube, therefore the coefficient of x_1 x_2 x_3 can be obtained by XORing up the functions of all eight of the vertices.
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This circuit diagram can be interpreted as a drawing of a hypergraph in which four vertices (depicted as white rectangles and disks) are connected by three hyperedges drawn as trees. Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs. In one possible visual representation for hypergraphs, similar to the standard graph drawing style in which curves in the plane are used to depict graph edges, a hypergraph's vertices are depicted as points, disks, or boxes, and its hyperedges are depicted as trees that have the vertices as their leaves... If the vertices are represented as points, the hyperedges may also be shown as smooth curves that connect sets of points, or as simple closed curves that enclose sets of points.... An order-4 Venn diagram, which can be interpreted as a subdivision drawing of a hypergraph with 15 vertices (the 15 colored regions) and 4 hyperedges (the 4 ellipses). In another style of hypergraph visualization, the subdivision model of hypergraph drawing,.
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In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs.
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They are polyhedral graphs, meaning that every Halin graph can be used to form the vertices and edges of a convex polyhedron, and the polyhedra formed from them have been called roofless polyhedra or domes. Every Halin graph has a Hamiltonian cycle through all its vertices, as well as cycles of almost all lengths up to their number of vertices. The Halin graphs can be recognized in linear time. Because Halin graphs have low treewidth, many computational problems that are hard on other kinds of planar graphs, such as finding Hamiltonian cycles, can also be solved quickly on Halin graphs.
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A graph is said to be -tough for a given real number if, for every integer , cannot be split into different connected components by the removal of fewer than vertices. For instance, a graph is -tough if the number of components formed by removing a set of vertices is always at most as large as the number of removed vertices. The toughness of a graph is the maximum for which it is -tough; this is a finite number for all finite graphs except the complete graphs, which by convention have infinite toughness. Graph toughness was first introduced by .
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For example, in image recognition applications, the results of image segmentation in image processing typically produces data graphs with the numbers of vertices much larger than in the model graphs data expected to match against. In the case of attributed graphs, even if the numbers of vertices and edges are the same, the matching still may be only inexact. Two categories of search methods are the ones based on identification of possible and impossible pairings of vertices between the two graphs and methods which formulate graph matching as an optimization problem.Graph-Based Methods in Computer Vision: Developments and Applications, p.
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The smaller polygon formed by removing this triangle has a 3-coloring by mathematical induction, and this coloring is easily extended to the one additional vertex of the removed triangle . Clearly, under a 3-coloring, every triangle must have all three colors. The vertices with any one color form a valid guard set, because every triangle of the polygon is guarded by its vertex with that color. Since the three colors partition the n vertices of the polygon, the color with the fewest vertices defines a valid guard set with at most \lfloor n/3\rfloor guards.
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A lattice is a set of orderly points that are connected by "edges". These points are called vertices and are connected to a certain number other vertices in the lattice by edges. The number of vertices each individual vertex is connected to is called the coordination number of the lattice, and it can be scaled up or down by changing the shape or dimension (2-dimensional to 3-dimensional, for example) of the lattice. This number is important in shaping the characteristics of the lattice protein because it controls the number of other residues allowed to be adjacent to a given residue.
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The algorithm proceeds according to the following steps. First, construct a single-vertex tree T by choosing (arbitrarily) one vertex. Then, while the tree T constructed so far does not yet include all of the vertices of the graph, let v be an arbitrary vertex that is not in T, perform a loop- erased random walk from v until reaching a vertex in T, and add the resulting path to T. Repeating this process until all vertices are included produces a uniformly distributed tree, regardless of the arbitrary choices of vertices at each step. A connection in the other direction is also true.
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The square of a graph In graph theory, a branch of mathematics, the kth power Gk of an undirected graph G is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in G is at most k. Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: G2 is called the square of G, G3 is called the cube of G, etc.. Graph powers should be distinguished from the products of a graph with itself, which (unlike powers) generally have many more vertices than the original graph.
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A vertex of a plane tiling or tessellation is a point where three or more tiles meet;M.V. Jaric, ed, Introduction to the Mathematics of Quasicrystals (Aperiodicity and Order, Vol 2) , Academic Press, 1989. generally, but not always, the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles. More generally, a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional faces.
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A graceful labelling; vertex labels are in black and edge labels in red A graph is known as graceful when its vertices are labeled from 0 to , the size of the graph, and this labelling induces an edge labelling from 1 to . For any edge e, the label of e is the positive difference between the two vertices incident with e. In other words, if e is incident with vertices labeled i and j, e will be labeled . Thus, a graph is graceful if and only if there exists an injection that induces a bijection from E to the positive integers up to .
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Such graphs are automatically symmetric, by definition. A t-arc is defined to be a sequence of t+1 vertices, such that any two consecutive vertices in the sequence are adjacent, and with any repeated vertices being more than 2 steps apart. A t-transitive graph is a graph such that the automorphism group acts transitively on t-arcs, but not on (t+1)-arcs. Since 1-arcs are simply edges, every symmetric graph of degree 3 or more must be t-transitive for some t, and the value of t can be used to further classify symmetric graphs.
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For instance, the 16-vertex planar graph shown in the illustration has edges. In this graph, there can be no perfect matching; for, if the center vertex is matched, the remaining unmatched vertices may be grouped into three different connected components with four, five, and five vertices, and the components with an odd number of vertices cannot be perfectly matched. However, the graph has maximum matchings with seven edges, so . Therefore, the number of colors needed to edge-color the graph is at least 24/7, and since the number of colors must be an integer it is at least four.
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Two greedy colorings of the same crown graph using different vertex orders. The right example generalises to 2-colorable graphs with vertices, where the greedy algorithm expends colors. In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not in general use the minimum number of colors possible.
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If such a graph exists, it would necessarily be a locally linear graph and a strongly regular graph with parameters (99,14,1,2). The first, third, and fourth parameters encode the statement of the problem: the graph should have 99 vertices, every pair of adjacent vertices should have 1 common neighbor, and every pair of non-adjacent vertices should have 2 common neighbors. The second parameter means that the graph is a regular graph with 14 edges per vertex. If this graph exists, it does not have any symmetries of order 11, which implies that its symmetries cannot take every vertex to every other vertex.
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Often, when an object is rigged with a skeleton, weights are applied to the vertices near the joints. The vertices closer to the joint will usually have a lower weight assigned; the reason is so that during deformation, the geometry of the skin does not fold in on itself. Weighting in this situation will most of the time be done automatically using skinning techniques but is often done by hand fine tune the skeleton's deformation effects. Another example of using weights with a skeleton structure would be to actually apply weights to vertices that are not part of a character's skin.
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A graph with eight vertices, and a tree decomposition of it onto a tree with six nodes. Each graph edge connects two vertices that are listed together at some tree node, and each graph vertex is listed at the nodes of a contiguous subtree of the tree. Each tree node lists at most three vertices, so the width of this decomposition is two. In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph.
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M2, M3 and M4 Mycielski graphs Applying the Mycielskian repeatedly, starting with the one-edge graph, produces a sequence of graphs Mi = μ(Mi−1), sometimes called the Mycielski graphs. The first few graphs in this sequence are the graph M2 = K2 with two vertices connected by an edge, the cycle graph M3 = C5, and the Grötzsch graph M4 with 11 vertices and 20 edges. In general, the graph Mi is triangle-free, (i−1)-vertex-connected, and i-chromatic. The number of vertices in Mi for i ≥ 2 is 3 × 2i−2 − 1 , while the number of edges for i = 2, 3, . . .
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For equal masses, one possible central configuration places the masses at the vertices of a regular polygon (forming a Klemperer rosette), a Platonic solid, or a regular polytope in higher dimensions. The centrality of the configuration follows from its symmetry. It is also possible to place an additional point, of arbitrary mass, at the center of mass of the system without changing its centrality. Placing three masses in an equilateral triangle, four at the vertices of a regular tetrahedron, or more generally masses at the vertices of a regular simplex produces a central configuration even when the masses are not equal.
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In a d-dimensional polytope with n=d+3 vertices, the linear Gale diagram consists of points on the unit circle (unit vectors) and at its center. The affine Gale diagram consists of labeled points or clusters of points on a line. Unlike for the case of n=d+3 vertices, it is not completely trivial to determine when two Gale diagrams represent the same polytope. Three-dimensional polyhedra with six vertices provide natural examples where the original polyhedron is of a low enough dimension to visualize, but where the Gale diagram still provides a dimension-reducing effect.
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A k-coloring, for some integer k, is an assignment of one of k colors to each vertex of a graph G such that the endpoints of each edge get different colors. The k-colorings of G correspond exactly to homomorphisms from G to the complete graph Kk. Indeed, the vertices of Kk correspond to the k colors, and two colors are adjacent as vertices of Kk if and only if they are different. Hence a function defines a homomorphism to Kk if and only if it maps adjacent vertices of G to different colors (i.e., it is a k-coloring).
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Every framed trace diagram corresponds to a multilinear function between tensor powers of the vector space V. The degree-1 vertices correspond to the inputs and outputs of the function, while the degree-n vertices correspond to the generalized Levi-Civita symbol (which is an anti-symmetric tensor related to the determinant). If a diagram has no output strands, its function maps tensor products to a scalar. If there are no degree-1 vertices, the diagram is said to be closed and its corresponding function may be identified with a scalar. By definition, a trace diagram's function is computed using signed graph coloring.
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A cyclic polygon (green), its midpoint polygon (red), and its midpoint- stretching polygon (pink) In geometry, the midpoint-stretching polygon of a cyclic polygon is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the circular arcs between the vertices of .. It may be derived from the midpoint polygon of (the polygon whose vertices are the edge midpoints) by placing the polygon in such a way that the circle's center coincides with the origin, and stretching or normalizing the vector representing each vertex of the midpoint polygon to make it have unit length.
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An enumeration of the vertices of a graph is said to be a LexBFS ordering if it is the possible output of the application of LexBFS to this graph. Let G=(V,E) be a graph with n vertices. Recall that N(v) is the set of neighbors of v. Let \sigma=(v_1,\dots,v_n) be an enumeration of the vertices of V. The enumeration \sigma is a LexBFS ordering (with source v_1) if, for all 1\le i with v_i\in N(v_j)\setminus N(v_k), there exists m such that v_m\in N(v_j)\setminus V(v_k).
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The straight skeleton of a polygon is defined by a continuous shrinking process in which the edges of the polygon are moved inwards parallel to themselves at a constant speed. As the edges move in this way, the vertices where pairs of edges meet also move, at speeds that depend on the angle of the vertex. If one of these moving vertices collides with a nonadjacent edge, the polygon is split in two by the collision, and the process continues in each part. The straight skeleton is the set of curves traced out by the moving vertices in this process.
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T-vertices created by joining a subdivided part with a non-subdivided part. T-vertices is a term used in computer graphics to describe a problem that can occur during mesh refinement or mesh simplification. The most common case occurs in naive implementations of continuous level of detail, where a finer- level mesh is "sewn" together with a coarser-level mesh by simply aligning the finer vertices on the edges of the coarse polygons. The result is a continuous mesh, however due to the nature of the z-buffer and certain lighting algorithms such as Gouraud shading, visual artifacts can often be detected.
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A pseudo-uniform polyhedron is a polyhedron which has regular polygons as faces and has the same vertex configuration at all vertices but is not vertex- transitive: it is not true that for any two vertices, there exists a symmetry of the polyhedron mapping the first isometrically onto the second. Thus, although all the vertices of a pseudo-uniform polyhedron appear the same, it is not isogonal. They are called pseudo-uniform polyhedra due to their resemblance to some true uniform polyhedra. There are two known pseudo-uniform polyhedra: the pseudorhombicuboctahedron and the pseudo-great rhombicuboctahedron.
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The results regarding the existence of the Napoleon points can be generalized in different ways. In defining the Napoleon points we begin with equilateral triangles drawn on the sides of the triangle ABC and then consider the centers X, Y, and Z of these triangles. These centers can be thought as the vertices of isosceles triangles erected on the sides of triangle ABC with the base angles equal to /6 (30 degrees). The generalizations seek to determine other triangles that, when erected over the sides of the triangle ABC, have concurrent lines joining their external vertices and the vertices of triangle ABC.
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Distinguishing coloring of a 4-hypercube graph In graph theory, a distinguishing coloring or distinguishing labeling of a graph is an assignment of colors or labels to the vertices of the graph that destroys all of the nontrivial symmetries of the graph. The coloring does not need to be a proper coloring: adjacent vertices are allowed to be given the same color. For the colored graph, there should not exist any one-to-one mapping of the vertices to themselves that preserves both adjacency and coloring. The minimum number of colors in a distinguishing coloring is called the distinguishing number of the graph.
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Shortest path (A, C, E, D, F) between vertices A and F in the weighted directed graph In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment.
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Also, it counts non-induced trees and bounded treewidth sub-graphs. This method is applied for sub-graphs of size up to 10. This algorithm counts the number of non-induced occurrences of a tree with vertices in a network with vertices as follows: # Color coding. Color each vertex of input network G independently and uniformly at random with one of the colors.
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Contracting the edge between the indicated vertices, resulting in graph G / {uv}. In graph theory, an edge contraction is an operation which removes an edge from a graph while simultaneously merging the two vertices that it previously joined. Edge contraction is a fundamental operation in the theory of graph minors. Vertex identification is a less restrictive form of this operation.
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A graph with connectivity 4. In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed. The vertex- connectivity, or just connectivity, of a graph is the largest k for which the graph is k-vertex-connected.
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Correlation functions are the sum of the connected Feynman diagrams, but the formalism treats the connected and disconnected diagrams differently. Internal lines end on vertices, while external lines go off to insertions. Introducing sources unifies the formalism, by making new vertices where one line can end. Sources are external fields, fields that contribute to the action, but are not dynamical variables.
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A simple polygonal chain A self-intersecting polygonal chain A closed polygonal chain In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain P is a curve specified by a sequence of points \scriptstyle(A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments connecting the consecutive vertices.
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The vertices of every triangle fall on a circle. (Because of this, some authors define "concyclic" only in the context of four or more points on a circle.). The circle containing the vertices of a triangle is called the circumscribed circle of the triangle. Several other sets of points defined from a triangle are also concyclic, with different circles; see nine-point circle.
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It is a trivial task when the convex polygon is specified in a traditional for polygons way, i.e., by the ordered sequence of its vertices v_1,\dots, v_m. When the input list of vertices (or edges) is unordered, the time complexity of the problems becomes O(m log m). A matching lower bound is known in the algebraic decision tree model of computation.
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Fix a set X of n vertices. For given r, an r-edge or block is a set of r vertices. A set of blocks is called a Turán (n,k,r) system (n ≥ k ≥ r) if every k-element subset of X contains a block. The Turán number T(n,k,r) is the minimum size of such a system.
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A circle with five chords and the corresponding circle graph. In graph theory, a circle graph is the intersection graph of a set of chords of a circle. That is, it is an undirected graph whose vertices can be associated with chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
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The concept of a triangulation may also be generalized somewhat to subdivisions into shapes related to triangles. In particular, a pseudotriangulation of a point set is a partition of the convex hull of the points into pseudotriangles, polygons that like triangles have exactly three convex vertices. As in point set triangulations, pseudotriangulations are required to have their vertices at the given input points.
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Another interpretation can represent this solid as a hexahedron, by considering pairs of trapezoids as a folded regular hexagon. It will have 6 faces (4 triangles, and 2 hexagons), 12 edges, and 8 vertices. It could also be seen as a folded tetrahedron also seeing pairs of end triangles as a folded rhombus. It would have 8 vertices, 10 edges, and 4 faces.
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An edge- graceful labeling of C_5 Consider the cycle with three vertices, C3. This is simply a triangle. One can label the edges 1, 2, and 3, and check directly that, along with the induced labeling on the vertices, this gives an edge- graceful labeling. Similar to paths, C_m is edge-graceful when m is odd and not when m is even.
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They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. Kn can be decomposed into trees Ti such that Ti has vertices.
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In graph theory, the Games graph is the largest known locally linear strongly regular graph. Its parameters as a strongly regular graph are (729,112,1,20). This means that it has 729 vertices, and 40824 edges (112 per vertex). Each edge is in a unique triangle (it is a locally linear graph) and each non- adjacent pair of vertices have exactly 20 shared neighbors.
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The smallest asymmetric non-trivial graphs have 6 vertices. The smallest asymmetric regular graphs have ten vertices; there exist ten-vertex asymmetric graphs that are 4-regular and 5-regular... One of the five smallest asymmetric cubic graphs is the twelve-vertex Frucht graph discovered in 1939.. According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs.
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Symmetries of a regular decagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edges. Gyration orders are given in the center. The regular decagon has Dih10 symmetry, order 20. There are 3 subgroup dihedral symmetries: Dih5, Dih2, and Dih1, and 4 cyclic group symmetries: Z10, Z5, Z2, and Z1.
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Conder, Malnič, Marušič, Pisanski and Potočnik rediscovered this 112-vertices graph in 2002 and named it the Ljubljana graph after the capital of Slovenia.Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; and Potočnik, P. "The Ljubljana Graph." 2002. . They proved that it was the unique 112-vertices edge- but not vertex-transitive cubic graph and therefore that was the graph found by Foster.
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Graph of an example equivalence with 7 classes An undirected graph may be associated to any symmetric relation on a set , where the vertices are the elements of , and two vertices and are joined if and only if . Among these graphs are the graphs of equivalence relations; they are characterized as the graphs such that the connected components are cliques.
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As such, the generalized Jemmis rule can be stated as follows: :The SEP requirement of condensed polyhedral clusters is m + n + o + p − q, where m is the number of subclusters, n is the number of vertices, o is the number of single-vertex shared condensations, p is the number of missing vertices and q is the number of caps.
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At high temperature and pressure, RbCl adopts the caesium chloride (CsCl) structure (NaCl and KCl undergo the same structural change at high pressures). Here, the chloride ions form a simple cubic arrangement with chloride anions occupying the vertices of a cube surrounding a central Rb+. This is RbCl's densest packing motif. Because a cube has eight vertices, both ions' coordination numbers equal eight.
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275-278) He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular hexacontagons. Only the g60 symmetry has no degrees of freedom but can seen as directed edges.
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The simplest qualitative construction of a dual polygon is a rectification operation, where the edges of a polygon are truncated down to vertices at the center of each original edge. New edges are formed between these new vertices. This construction is not reversible. That is, the polygon generated by applying it twice is in general not similar to the original polygon.
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A vertex cover in a graph is a set of vertices that touches every edge in the graph. A vertex cover is minimal, or irredundant, if removing any vertex from it would destroy the covering property. It is minimum if there is no other vertex cover with fewer vertices. A well-covered graph is one in which every minimal cover is also minimum.
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Symmetries of a regular tetradecagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center. The regular tetradecagon has Dih14 symmetry, order 28. There are 3 subgroup dihedral symmetries: Dih7, Dih2, and Dih1, and 4 cyclic group symmetries: Z14, Z7, Z2, and Z1.
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In other words, the group that acts on the children of a node is the symmetric group S3. We define the weight of such a ternary tree to be the number of nodes (or non-leaf vertices). Rooted ternary trees on 0, 1, 2, 3 and 4 nodes (=non-leaf vertices). The root is shown in blue, the leaves are not shown.
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275-278) He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular enneacontagons. Only the g90 symmetry has no degrees of freedom but can seen as directed edges.
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The patterns aim at identifying meanings using the local structural properties of the co- occurrence graph. A randomized algorithm which partitions the graph vertices by iteratively transferring the mainstream message (i.e. word sense) to neighboring vertices is Chinese Whispers. By applying co-occurrence graphs approaches have been shown to achieve the state-of-the-art performance in standard evaluation tasks.
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The 12 vertices of the desmic system and the 16 lines formed in this way are the points and lines of a Reye configuration.
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Figure 1. Example of a Fibonacci heap. It has three trees of degrees 0, 1 and 3. Three vertices are marked (shown in blue).
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Therefore, the surface area is exactly (2n-4)\pi. In an ideal polyhedron, all face angles and all solid angles at vertices are zero.
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All but three of the vertices of the Herschel graph have degree three. Tait's conjecture. Reprinted in Scientific Papers, Vol. II, pp. 85–98.
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A planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect.
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Its 30 vertices represent the root vectors of the simple Lie group A5. It is also the vertex figure of the 5-simplex honeycomb.
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In graphs that have negative cycles, the set of shortest simple paths from v to all other vertices do not necessarily form a tree.
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Some authors define cliques in a way that requires them to be maximal, and use other terminology for complete subgraphs that are not maximal. A maximum clique of a graph, G, is a clique, such that there is no clique with more vertices. Moreover, the clique number ω(G) of a graph G is the number of vertices in a maximum clique in G. The intersection number of G is the smallest number of cliques that together cover all edges of G. The clique cover number of a graph G is the smallest number of cliques of G whose union covers the set of vertices V of the graph. A maximum clique transversal of a graph is a subset of vertices with the property that each maximum clique of the graph contains at least one vertex in the subset.
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No matter which choice one makes of a color from the list of A and a color from the list of B, there will be some other vertex such that both of its choices are already used to color its neighbors. Thus, G is not 2-choosable. On the other hand, it is easy to see that G is 3-choosable: picking arbitrary colors for the vertices A and B leaves at least one available color for each of the remaining vertices, and these colors may be chosen arbitrarily. A list coloring instance on the complete bipartite graph K3,27 with three colors per vertex. No matter which colors are chosen for the three central vertices, one of the outer 27 vertices will be uncolorable, showing that the list chromatic number of K3,27 is at least four.
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A linear hypergraph (also known as partial linear space) is a hypergraph with the property that every two hyperedges have at most one vertex in common. A hypergraph is said to be uniform if all of its hyperedges have the same number of vertices as each other. The cliques of size in the Erdős–Faber–Lovász conjecture may be interpreted as the hyperedges of an -uniform linear hypergraph that has the same vertices as the underlying graph. In this language, the Erdős–Faber–Lovász conjecture states that, given any -uniform linear hypergraph with hyperedges, one may -color the vertices such that each hyperedge has one vertex of each color.. A simple hypergraph is a hypergraph in which at most one hyperedge connects any pair of vertices and there are no hyperedges of size at most one.
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Perkel graphs with 19-fold symmetry The vertices and edges form the Perkel graph, the unique distance-regular graph with intersection array {6,5,2;1,1,3}, discovered by .
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The other three polyhedra with this property are the regular octahedron, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces..
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In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.
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If there is a rainbow shortest path between each pair of vertices, the graph is said to be strongly rainbow-connected (or strongly rainbow colored).
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In probability theory, a random recursive tree is a rooted tree chosen uniformly at random from the recursive trees with a given number of vertices.
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For instance a bond graph connecting two vertices by k edges has k different spanning trees, each consisting of a single one of these edges.
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This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.
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The pentagram was said to have been so called from Pythagoras himself having written the letters Υ, Γ, Ι, Θ (= /ei/), Α on its vertices.
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In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices.
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A tetrahedron has 4 vertices, 6 edges, and is bounded by 4 triangular faces. In most cases a tetrahedral volume mesh can be generated automatically.
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The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended).
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The Fibonacci cube may be defined in terms of Fibonacci codes and Hamming distance, independent sets of vertices in path graphs, or via distributive lattices.
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Therefore, every convex combination of points of X belongs to a simplex whose vertices belong to X, and the third and fourth definitions are equivalent.
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This algorithm classifies vertices against the given line in the implicit form p: ax + by + c = 0. As the polygon is assumed to be convex and vertices are ordered clockwise or anti-clockwise, binary search can be applied and leads to a O(lg N) run- time complexity.Skala, V.: O(lg N) Line Clipping Algorithm in E2, Computers & Graphics, Pergamon Press, Vol. 18, No. 4, 1994.
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Every distance-hereditary graph is also a parity graph, a graph in which every two induced paths between the same pair of vertices both have odd length or both have even length., p.45. Every even power of a distance-hereditary graph G (that is, the graph G2i formed by connecting pairs of vertices at distance at most 2i in G) is a chordal graph., Theorem 10.6.
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On 8 April 2018, de Grey posted a paper to the arXiv explicitly constructing a unit-distance graph in the plane that cannot be colored with fewer than five colors. The previous lower bound is due to the problem's original proposal in 1950 by Hugo Hadwiger and Edward Nelson. De Grey's graph has 1581-vertices but it has since been reduced to 633 vertices by independent researchers.
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It is also possible for two vertices to exist at the same spatial coordinates, or two faces to exist at the same location. Situations such as these are usually not desired and many packages support an auto-cleanup function. If auto-cleanup is not present, however, they must be deleted manually. A group of polygons which are connected by shared vertices is referred to as a mesh.
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In mathematics, dependent random choice is a simple yet powerful probabilistic technique which shows how to find a large set of vertices in a dense graph such that every small subset of vertices has a lot of common neighbors. It is a useful tool to embed a graph into another graph with many edges, and thus has its application in extremal graph theory and Ramsey theory.
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In the mathematical discipline of graph theory, the (m,n)-lollipop graph is a special type of graph consisting of a complete graph (clique) on m vertices and a path graph on n vertices, connected with a bridge. The special case of the (2n/3,n/3)-lollipop graphs are known as graphs which achieve the maximum possible hitting time, cover time and commute time.
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If there is a perfect matching, then both the matching number and the edge cover number equal . A perfect matching can only occur when the graph has an even number of vertices. A near-perfect matching is one in which exactly one vertex is unmatched. This can only occur when the graph has an odd number of vertices, and such a matching must be maximum.
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The number of vertices in a proprism is equal to the product of the number of vertices in all the polytopes in the product. The minimum symmetry order of a proprism is the product of the symmetry orders of all the polytopes. A higher symmetry order is possible if polytopes in the product are identical. A proprism is convex if all its product polytopes are convex.
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Newman, Mark E. J. Networks: an Introduction. Oxford: Oxford UP, 2010. p.168 Figure 6: A connected directed network with two components (shaded) :Eigenvector centrality is an extension of the concept of degree centrality, based on the fact that in many networks not all vertices have the same weight or importance. A vertex's importance in its network increases if it has more connections to important vertices.
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For every H, u(H)\leq \tau(H), since every cover must contain at least one point from each edge in any matching. If H is r-uniform (each hyperedge has exactly r vertices), then \tau(H) \leq r\cdot u(H), since the union of the edges from any maximal matching is a set of at most rv vertices that meets every edge.
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G4 A gear graph, denoted Gn is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, Gn has 2n+1 vertices and 3n edges. Gear graphs are examples of squaregraphs, and play a key role in the forbidden graph characterization of squaregraphs. Gear graphs are also known as cogwheels and bipartite wheels.
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Chvátal's art gallery theorem, named after Václav Chvátal, gives an upper bound on the minimal number of guards. It states that \left\lfloor n/3 \right\rfloor guards are always sufficient and sometimes necessary to guard a simple polygon with n vertices. The question about how many vertices/watchmen/guards were needed was posed to Chvátal by Victor Klee in 1973. Chvátal proved it shortly thereafter.
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A 3-coloring of the vertices of a triangulated polygon. The blue vertices form a set of three guards, as few as is guaranteed by the art gallery theorem. However, this set is not optimal: the same polygon can be guarded by only two guards. Steve Fisk's proof is so short and elegant that it was chosen for inclusion in Proofs from THE BOOK.
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Circular layout of the Chvátal graph Circular layout of a state diagram for the border gateway protocol Incremental construction of a circular layout for the Barabási–Albert model of social network formation In graph drawing, a circular layout is a style of drawing that places the vertices of a graph on a circle, often evenly spaced so that they form the vertices of a regular polygon.
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Symmetries of a regular icosagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center. The regular icosagon has Dih20 symmetry, order 40. There are 5 subgroup dihedral symmetries: (Dih10, Dih5), and (Dih4, Dih2, and Dih1), and 6 cyclic group symmetries: (Z20, Z10, Z5), and (Z4, Z2, Z1).
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A graph with six vertices. Many trajectory inference algorithms use graphs to build the trajectory. Many methods represent the structure of the dynamic process via a graph-based approach. In such an approach the vertices of the graph correspond to states in the dynamic process, such as cell types in cell differentiation, and the edges between the nodes correspond to transitions between the states.
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Therefore, there are k\times(k-\lambda-1) edges between Level 1 and Level 2. # Vertices in Level 2 are not directly connected to the root, hence they must have μ common neighbors with the root, and these common neighbors must all be in Level 1. There are (v-k-1) vertices in Level 2, and each is connected to μ nodes in Level 1.
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In graph theory, an induced matching or strong matching is a subset of the edges of an undirected graph that do not share any vertices (it is a matching) and includes every edge connecting any two vertices in the subset (it is an induced subgraph). An induced matching can also be described as an independent set in the square of the line graph of the given graph.
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Research performed in 2004 used electron microscopy to predict that UL-6 forms 11, 12, 13, and 14-unit polymers. The dodecameric form was found to be most likely. Refinements to the electron microscopy in 2007 allowed finding that the portal is a twelve (12)-unit polymer present at one of the twelve capsid vertices instead of the UL-19 pentamer found at non- portal vertices.
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In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non- compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by . The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3.
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Due to the huge amount of gene data that can be collected, searching for pathways or motifs can be highly time consuming. However, by exploiting the color-coding method, the motifs or signaling pathways with k=O(\log n) vertices in a network with vertices can be found very efficiently in polynomial time. Thus, this enables us to explore more complex or larger structures in PPI networks.
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In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by . The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3.
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An empty graph is a graph that has an empty set of vertices (and thus an empty set of edges). The order of a graph is its number of vertices . The size of a graph is its number of edges . However, in some contexts, such as for expressing the computational complexity of algorithms, the size is (otherwise, a non-empty graph could have a size 0).
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The octahedron, a 3-cross polytope whose edges and vertices form K2,2,2, a Turán graph T(6,3). Unconnected vertices are given the same color in this face-centered projection. Several choices of the parameter r in a Turán graph lead to notable graphs that have been independently studied. The Turán graph T(2n,n) can be formed by removing a perfect matching from a complete graph K2n.
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In 1973, Stefan Burr and Paul Erdős made the following conjecture: :For every integer p there exists a constant cp so that any p-degenerate graph G on n vertices has Ramsey number at most cp n. That is, if an n-vertex graph G is p-degenerate, then a monochromatic copy of G should exist in every two-edge-colored complete graph on cp n vertices.
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It gives a sufficient condition for a graph to be Hamiltonian, essentially stating that a graph with sufficiently many edges must contain a Hamilton cycle. Specifically, the theorem considers the sum of the degrees of pairs of non-adjacent vertices: if every such pair has a sum that at least equals the total number of vertices in the graph, then the graph is Hamiltonian.
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The Dürer graph G(6, 2). In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter.
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He introduced a formula governing the relationship between the number of edges, vertices, and faces of a convex polyhedron. Given such a polyhedron, the alternating sum of vertices, edges and faces equals a constant: V − E + F = 2\. This constant, χ, is the Euler characteristic of the plane. The study and generalization of this equation, specially by Cauchy and Lhuillier, is at the origin of topology.
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In addition, for each edge vivj of G, the Mycielski graph includes two edges, uivj and viuj. Thus, if G has n vertices and m edges, μ(G) has 2n+1 vertices and 3m+n edges. The only new triangles in μ(G) are of the form vivjuk, where vivjvk is a triangle in G. Thus, if G is triangle-free, so is μ(G).
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The Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point. In either its simple form or its self-intersecting form, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.
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Kempe chains in the Errera graph. The four color theorem states that the vertices of every planar graph can be colored with four colors, so that no two adjacent vertices have equal colors. An erroneous proof was published in 1879 by Alfred Kempe, but it was discovered to be erroneous by 1890. The four color theorem was not given a valid proof until 1976.
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The same basic algorithm can be applied to triangular meshes, which consist of connected triangles with data assigned to the vertices. For example, a scattered set of data points could be connected with a Delaunay triangulation to allow the data field to be contoured. A triangular cell is always planar, because it is a 2-simplex (i.e. specified by n+1 vertices in an n-dimensional space).
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The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length \sqrt 3, while AC (shown in red) is a face diagonal and has length \sqrt 2. In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal.
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The Goldner–Harary graph, an example of a planar 3-tree. In graph theory, a k-tree is an undirected graph formed by starting with a (k + 1)-vertex complete graph and then repeatedly adding vertices in such a way that each added vertex v has exactly k neighbors U such that, together, the k + 1 vertices formed by v and U form a clique.
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One way of doing this is by minimum weight matching using algorithms of O(n^3). Creating a matching Making a graph into an Eulerian graph starts with the minimum spanning tree. Then all the vertices of odd order must be made even. So a matching for the odd degree vertices must be added which increases the order of every odd degree vertex by one.
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Because these graphs are bipartite and have Hamiltonian paths, their maximum independent sets have a number of vertices that is equal to half of the number of vertices in the whole graph, rounded up to the nearest integer., p.6. The diameter of a Fibonacci cube of order n is n, and its radius is n/2 (again, rounded up to the nearest integer)., p.9.
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A graph with three components. In graph theory, a component, sometimes called a connected component, of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph. For example, the graph shown in the illustration has three components. A vertex with no incident edges is itself a component.
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The above interactions show some basic interaction vertices – Feynman diagrams in the standard model are built from these vertices. Higgs boson interactions are however not shown, and neutrino oscillations are commonly added. The charge of the W bosons are dictated by the fermions they interact with. We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model Lagrangian density.
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Figure 3. Face-vertex meshes Face-vertex meshes represent an object as a set of faces and a set of vertices. This is the most widely used mesh representation, being the input typically accepted by modern graphics hardware. Face-vertex meshes improve on VV-mesh for modeling in that they allow explicit lookup of the vertices of a face, and the faces surrounding a vertex.
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Computer-aided design (CAD) programs use specular highlights as visual cues to convey a sense of surface curvature when rendering 3D objects. However, many CAD programs exhibit problems in sampling specular highlights because the specular lighting computations are only performed at the vertices of the mesh used to represent the object, and interpolation is used to estimate lighting across the surface of the object. Problems occur when the mesh vertices are not dense enough, resulting in insufficient sampling of the specular lighting. This in turn results in highlights with brightness proportionate to the distance from mesh vertices, ultimately compromising the visual cues that indicate curvature.
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Every directed path graph with an even number of vertices is skew-symmetric, via a symmetry that swaps the two ends of the path. However, path graphs with an odd number of vertices are not skew-symmetric, because the orientation-reversing symmetry of these graphs maps the center vertex of the path to itself, something that is not allowed for skew-symmetric graphs. Similarly, a directed cycle graph is skew- symmetric if and only if it has an even number of vertices. In this case, the number of different mappings σ that realize the skew symmetry of the graph equals half the length of the cycle.
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A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex v, v and the neighbors of v that occur after v in the order form a clique. A graph is chordal if and only if it has a perfect elimination ordering. (see also ) show that a perfect elimination ordering of a chordal graph may be found efficiently using an algorithm known as lexicographic breadth-first search. This algorithm maintains a partition of the vertices of the graph into a sequence of sets; initially this sequence consists of a single set with all vertices.
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A cycle (or a circuit) in a hypergraph is a cyclic alternating sequence of distinct vertices and hyperedges: (v1, e1, v2, e2, ..., vk, ek, vk+1=v1), where every vertex vi is contained in both ei−1 and ei. The number k is called the length of the cycle. A hypergraph is balanced iff every odd-length cycle C in H has an edge containing at least three vertices of C. Note that in a simple graph all edges contain only two vertices. Hence, a simple graph is balanced iff it contains no odd-length cycles at all, which holds iff it is bipartite.
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Let G be a directed graph, S be a set of starting vertices, and T be a set of destination vertices (not necessarily disjoint from S). The gammoid \Gamma derived from this data has T as its set of elements. A subset I of T is independent in \Gamma if there exists a set of vertex-disjoint paths whose starting points all belong to S and whose ending points are exactly I.. A strict gammoid is a gammoid in which the set T of destination vertices consists of every vertex in G. Thus, a gammoid is a restriction of a strict gammoid, to a subset of its elements.
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Every Halin graph is 3-connected, meaning that it is not possible to delete two vertices from it and disconnect the remaining vertices. It is edge-minimal 3-connected, meaning that if any one of its edges is removed, the remaining graph will no longer be 3-connected. By Steinitz's theorem, as a 3-connected planar graph, it can be represented as the set of vertices and edges of a convex polyhedron; that is, it is a polyhedral graph. And, as with every polyhedral graph, its planar embedding is unique up to the choice of which of its faces is to be the outer face.
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Suppose that the vertices of the polygon P are given by P_1,P_3,P_5,\ldots The image of P under the pentagram map is the polygon Q with vertices Q_2,Q_4,Q_6,\ldots as shown in the figure. Here Q_4 is the intersection of the diagonals (P_1P_5) and (P_3P_7) , and so on. test On a basic level, one can think of the pentagram map as an operation defined on convex polygons in the plane. From a more sophisticated point of view, the pentagram map is defined for a polygon contained in the projective plane over a field provided that the vertices are in sufficiently general position.
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Its representation is not so simple as in the planar case, however. In higher dimensions, even if the vertices of a convex polytope are known, construction of its faces is a non-trivial task, as is the dual problem of constructing the vertices given the faces. The size of the output face information may be exponentially larger than the size of the input vertices, and even in cases where the input and output are both of comparable size the known algorithms for high-dimensional convex hulls are not output-sensitive due both to issues with degenerate inputs and with intermediate results of high complexity..
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The criss-cross algorithm was used in an algorithm for enumerating all the vertices of a polytope, which was published by David Avis and Komei Fukuda in 1992. Avis and Fukuda presented an algorithm which finds the v vertices of a polyhedron defined by a nondegenerate system of n linear inequalities in D dimensions (or, dually, the v facets of the convex hull of n points in D dimensions, where each facet contains exactly D given points) in time O(nDv) and O(nD) space.The v vertices in a simple arrangement of n hyperplanes in D dimensions can be found in O(n2Dv) time and O(nD) space complexity.
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The Herschel graph is a planar graph: it can be drawn in the plane with none of its edges crossing. It is also 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph. It is a bipartite graph: its vertices can be separated into two subsets of five and six vertices respectively, such that every edge has an endpoint in each subset (the red and blue subsets in the picture). As with any bipartite graph, the Herschel graph is a perfect graph : the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph.
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Every complete graph with an odd number n of vertices has a Hamiltonian decomposition. This result, which is a special case of the Oberwolfach problem of decomposing complete graphs into isomorphic 2-factors, was attributed to Walecki by Édouard Lucas in 1892. Walecki's construction places n-1 of the vertices into a regular polygon, and covers the complete graph in this subset of vertices with (n-1)/2 Hamiltonian paths that zigzag across the polygon, with each path rotated from each other path by a multiple of \pi/(n-1). The paths can then all be completed to Hamiltonian cycles by connecting their ends through the remaining vertex.
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According to Brooks' theorem every connected cubic graph other than the complete graph K4 can be colored with at most three colors. Therefore, every connected cubic graph other than K4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices. According to Vizing's theorem every cubic graph needs either three or four colors for an edge coloring. A 3-edge-coloring is known as a Tait coloring, and forms a partition of the edges of the graph into three perfect matchings.
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A zero in these locations will be incorrectly interpreted as an edge with no distance, cost, etc. If W is an n \times n matrix containing the edge weights of a graph, then W^k (using this distance product) gives the distances between vertices using paths of length at most k edges, and W^n is the distance matrix of the graph. An arbitrary graph on vertices can be modeled as a weighted complete graph on vertices by assigning a weight of one to each edge of the complete graph that corresponds to an edge of and zero to all other edges. for this complete graph is the adjacency matrix of .
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In the monadic second-order logic of graphs, the variables represent objects of up to four types: vertices, edges, sets of vertices, and sets of edges. There are two main variations of monadic second-order graph logic: MSO1 in which only vertex and vertex set variables are allowed, and MSO2 in which all four types of variables are allowed. The predicates on these variables include equality testing, membership testing, and either vertex-edge incidence (if both vertex and edge variables are allowed) or adjacency between pairs of vertices (if only vertex variables are allowed). Additional variations in the definition allow additional predicates such as modular counting predicates.
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The Petersen graph is a (cubic) symmetric graph. Any pair of adjacent vertices can be mapped to another by an automorphism, since any five-vertex ring can be mapped to any other. In the mathematical field of graph theory, a graph G is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphism :f : V(G) → V(G) such that :f(u1) = u2 and f(v1) = v2. In other words, a graph is symmetric if its automorphism group acts transitively upon ordered pairs of adjacent vertices (that is, upon edges considered as having a direction).
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Switching equivalence means that two graphs are related by switching, and an equivalence class of signed graphs under switching is called a switching class. Sometimes these terms are applied to equivalence of signed graphs under the combination of switching and isomorphism, especially when the graphs are unlabeled; but to distinguish the two concepts the combined equivalence may be called switching isomorphism and an equivalence class under switching isomorphism may be called a switching isomorphism class. Switching a set of vertices affects the adjacency matrix by negating the rows and columns of the switched vertices. It affects the incidence matrix by negating the rows of the switched vertices.
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A 9-vertex graph in which every edge belongs to a unique triangle and every non-edge is the diagonal of a unique quadrilateral. The 99-graph problem asks for a 99-vertex graph with the same property. In graph theory, Conway's 99-graph problem is an unsolved problem asking whether there exists an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors. Equivalently, every edge should be part of a unique triangle and every non-adjacent pair should be one of the two diagonals of a unique 4-cycle.
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A directed graph with three vertices (blue circles) and three edges (black arrows). In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. These pairs are known as edges (also called links or lines), and for a directed graph are also known as arrows.
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The graph formed by the edges and vertices of the dual polyhedron is its dual graph. More generally, for any polyhedron whose faces form a closed surface, the vertices and edges of the polyhedron form a graph embedded on this surface, and the vertices and edges of the (abstract) dual polyhedron form the dual graph. An abstract polyhedron is a certain kind of partially ordered set (poset) of elements, such that adjacencies, or connections, between elements of the set correspond to adjacencies between elements (faces, edges, etc.) of a polyhedron. Every such poset has a dual poset, formed by reversing all of the order relations.
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A beziergon (also called bezigon) is a closed path composed of Bézier curves. It is similar to a polygon in that it connects a set of vertices by lines, but whereas in polygons the vertices are connected by straight lines, in a beziergon the vertices are connected by Bézier curves.Microsoft polybezier API Papyrus beziergon API reference "A better box of crayons". InfoWorld. 1991. Some authors even call a C0 composite Bézier curve a "Bézier spline"; the latter term is however used by other authors as a synonym for the (non-composite) Bézier curve, and they add "composite" in front of "Bézier spline" to denote the composite case.
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It is automatically true when the graph contains an odd cycle, because the independent set of all heavy vertices cannot cover all the edges of the cycle. Therefore, the more interesting case of the conjecture is for bipartite graphs, which have no odd cycles. Another equivalent formulation of the conjecture is that, in every bipartite graph, there exist two vertices, one on each side of the bipartition, such that each of these two vertices belongs to at most half of the graph's maximal independent sets. This conjecture is known to hold for chordal bipartite graphs, bipartite series-parallel graphs, and bipartite graphs of maximum degree three.
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From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle. The Gergonne triangle or intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle.
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The Prüfer sequence of a labeled tree on n vertices is a unique sequence of length n − 2 on the labels 1 to n. For a given sequence S of length n-2 on the labels 1 to n, there is a unique labeled tree whose Prüfer sequence is S. The immediate consequence is that Prüfer sequences provide a bijection between the set of labeled trees on n vertices and the set of sequences of length n − 2 on the labels 1 to n. The latter set has size nn−2, so the existence of this bijection proves Cayley's formula, i.e. that there are nn−2 labeled trees on n vertices.
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A permutation set of an n-by-n matrix X is a set of n entries of X containing exactly one entry from each row and from each column. A theorem by Dénes Kőnig says that:. > Every bistochastic matrix has a permutation-set in which all entries are > positive. The positivity graph of an n-by-n matrix X is a bipartite graph with 2n vertices, in which the vertices on one side are n rows and the vertices on the other side are the n columns, and there is an edge between a row and a column iff the entry at that row and column is positive.
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General homomorphisms can also be thought of as a kind of coloring: if the vertices of a fixed graph H are the available colors and edges of H describe which colors are compatible, then an H-coloring of G is an assignment of colors to vertices of G such that adjacent vertices get compatible colors. Many notions of graph coloring fit into this pattern and can be expressed as graph homomorphisms into different families of graphs. Circular colorings can be defined using homomorphisms into circular complete graphs, refining the usual notion of colorings. Fractional and b-fold coloring can be defined using homomorphisms into Kneser graphs.
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The all-pairs widest path problem has applications in the Schulze method for choosing a winner in multiway elections in which voters rank the candidates in preference order. The Schulze method constructs a complete directed graph in which the vertices represent the candidates and every two vertices are connected by an edge. Each edge is directed from the winner to the loser of a pairwise contest between the two candidates it connects, and is labeled with the margin of victory of that contest. Then the method computes widest paths between all pairs of vertices, and the winner is the candidate whose vertex has wider paths to each opponent than vice versa.
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Winged-edge meshes are not the only representation which allows for dynamic changes to geometry. A new representation which combines winged-edge meshes and face-vertex meshes is the render dynamic mesh, which explicitly stores both, the vertices of a face and faces of a vertex (like FV meshes), and the faces and vertices of an edge (like winged-edge). Render dynamic meshes require slightly less storage space than standard winged-edge meshes, and can be directly rendered by graphics hardware since the face list contains an index of vertices. In addition, traversal from vertex to face is explicit (constant time), as is from face to vertex.
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The triangle where two vertices are ideal points and the remaining angle is right, one of the first hyperbolic triangles (1818) described by Ferdinand Karl Schweikart.
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A graph colouring is a subclass of graph labellings. Vertex colourings assign different labels to adjacent vertices, while edge colourings assign different labels to adjacent edges.
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In graph theory, the Poussin graph is a planar graph with 15 vertices and 39 edges. It is named after Charles Jean de la Vallée-Poussin.
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Betweenness centrality is related to a network's connectivity, in so much as high betweenness vertices have the potential to disconnect graphs if removed (see cut set) .
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The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon.
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The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±1,±1,±1,±3) with an odd number of plus signs.
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The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±3,±3,±3,±5) with an odd number of plus signs.
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The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±1,±3,±3,±5) with an odd number of plus signs.
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The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±3,±3,±5,±7) with an odd number of plus signs.
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The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±1,±1,±3,±5) with an odd number of plus signs.
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The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±3,±3,±5,±7) with an odd number of plus signs.
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The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±1,±3,±5,±7) with an odd number of plus signs.
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The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±3,±3,±5,±7) with an odd number of plus signs.
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For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces.
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The Gosset graph, named after Thorold Gosset, is a specific regular graph (1-skeleton of the 7-dimensional 321 polytope) with 56 vertices and valency 27..
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In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of O(√n) vertices from an n-vertex graph (where the O invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most 2n/3 vertices. A weaker form of the separator theorem with O(√n log n) vertices in the separator instead of O(√n) was originally proven by , and the form with the tight asymptotic bound on the separator size was first proven by . Since their work, the separator theorem has been reproven in several different ways, the constant in the O(√n) term of the theorem has been improved, and it has been extended to certain classes of nonplanar graphs.
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Sumner's Universal Tournament Conjecture, Douglas B. West, retrieved 2010-09-17. A family of graphs has a universal graph of polynomial size, containing every -vertex graph as an induced subgraph, if and only if it has an adjacency labelling scheme in which vertices may be labeled by -bit bitstrings such that an algorithm can determine whether two vertices are adjacent by examining their labels. For, if a universal graph of this type exists, the vertices of any graph in may be labeled by the identities of the corresponding vertices in the universal graph, and conversely if a labeling scheme exists then a universal graph may be constructed having a vertex for every possible label.. In older mathematical terminology, the phrase "universal graph" was sometimes used to denote a complete graph. The notion of universal graph has been adapted and used for solving mean payoff games.
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On the other hand, the 1-planar graphs, which are not closed under minors, have also bounded book thickness,. but some 1-planar graphs including have book thickness at least four.. Every shallow minor of a graph of bounded book thickness is a sparse graph, whose ratio of edges to vertices is bounded by a constant that depends only on the depth of the minor and on the book thickness. That is, in the terminology of , the graphs of bounded book thickness have bounded expansion. However, even the graphs of bounded degree, a much stronger requirement than having bounded expansion, can have unbounded book thickness.. Because graphs of book thickness two are planar graphs, they obey the planar separator theorem: they have separators, subsets of vertices whose removal splits the graph into pieces with at most vertices each, with only O(\sqrt n) vertices in the separator.
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By allowing the same vertices to be reached by multiple paths, a DAFSA may use significantly fewer vertices than the strongly related trie data structure. Consider, for example, the four English words "tap", "taps", "top", and "tops". A trie for those four words would have 12 vertices, one for each of the strings formed as a prefix of one of these words, or for one of the words followed by the end-of-string marker. However, a DAFSA can represent these same four words using only six vertices vi for 0 ≤ i ≤ 5, and the following edges: an edge from v0 to v1 labeled "t", two edges from v1 to v2 labeled "a" and "o", an edge from v2 to v3 labeled "p", an edge v3 to v4 labeled "s", and edges from v3 and v4 to v5 labeled with the end-of- string marker.
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An abstract polytope is a partially ordered set P (whose elements are called faces) with properties modeling those of the inclusions of faces of convex polytopes. The rank (or dimension) of an abstract polytope is determined by the length of the maximal ordered chains of its faces, and an abstract polytope of rank n is called an abstract n-polytope. For abstract polytopes of rank 2, this means that the elements of the partially ordered set are sets of vertices with either zero vertices (the empty set), one vertex, two vertices (an edge), or the entire vertex set, ordered by inclusion of sets, that each vertex belongs to exactly two edges, and that the undirected graph formed by the vertices and edges is connected. An abstract polytope is called an abstract apeirotope if it has infinitely many elements, and an abstract 2-apeirotope is called an abstract apeirogon.
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The squaregraphs include as special cases trees, grid graphs, gear graphs, and the graphs of polyominos. As well as being planar graphs, squaregraphs are median graphs, meaning that for every three vertices u, v, and w there is a unique median vertex m(u,v,w) that lies on shortest paths between each pair of the three vertices.. See for a discussion of planar median graphs more generally. As with median graphs more generally, squaregraphs are also partial cubes: their vertices can be labeled with binary strings such that the Hamming distance between strings is equal to the shortest path distance between vertices. The graph obtained from a squaregraph by making a vertex for each zone (an equivalence class of parallel edges of quadrilaterals) and an edge for each two zones that meet in a quadrilateral is a circle graph determined by a triangle-free chord diagram of the unit disk.
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In one direction, suppose one is given as input a graph G, and must test whether G is triangle-free. From G, construct a new graph H having as vertices each set of zero, one, or two adjacent vertices of G. Two such sets are adjacent in H when they differ by exactly one vertex. An equivalent description of H is that it is formed by splitting each edge of G into a path of two edges, and adding a new vertex connected to all the original vertices of G. This graph H is by construction a partial cube, but it is a median graph only when G is triangle-free: if a, b, and c form a triangle in G, then {a,b}, {a,c}, and {b,c} have no median in H, for such a median would have to correspond to the set {a,b,c}, but sets of three or more vertices of G do not form vertices in H. Therefore, G is triangle-free if and only if H is a median graph. In the case that G is triangle-free, H is its simplex graph.
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The snub disphenoid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well- covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the pentagonal bipyramid, and an irregular polyhedron with 12 vertices and 20 triangular faces.. The snub disphenoid has the same symmetries as a tetragonal disphenoid: it has an axis of 180° rotational symmetry through the midpoints of its two opposite edges, two perpendicular planes of reflection symmetry through this axis, and four additional symmetry operations given by a reflection perpendicular to the axis followed by a quarter-turn and possibly another reflection parallel to the axis.. That is, it has antiprismatic symmetry, a symmetry group of order 8\. Spheres centered at the vertices of the snub disphenoid form a cluster that, according to numerical experiments, has the minimum possible Lennard-Jones potential among all eight-sphere clusters.. Up to symmetries and parallel translation, the snub disphenoid has five types of simple (non-self-crossing) closed geodesics.
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The triangle where all vertices are ideal points, an ideal triangle is the largest possible triangle in hyperbolic geometry because of the zero sum of the angles.
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Then Tucker's lemma states that T contains a complementary edge - an edge (a 1-simplex) whose vertices are labelled by the same number but with opposite signs.
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The wheel graph Wn is a graph on n vertices constructed by connecting a single vertex to every vertex in an (n − 1)-cycle. Wheels W_4 – W_9.
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Since for every tree , we can easily count the number of trees that are within a forest by subtracting the difference between total vertices and total edges. .
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Cartesian coordinates for the vertices of this compound are all the permutations of : (±1, 0, ±τ) where τ = (1+)/2 is the golden ratio (sometimes written φ).
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Simpath is an algorithm introduced by Donald Knuth that constructs a zero- suppressed decision diagram (ZDD) representing all simple paths between two vertices in a given graph.
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Its 20 vertices represent the root vectors of the simple Lie group A4. It is also the vertex figure for the 5-cell honeycomb in 4-space.
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150px Orthographic projection of 10-simplex with 11 vertices, 55 edges. The abstract 11-cell contains the same number of vertices and edges as the 10-dimensional 10-simplex, and contains 1/3 of its 165 faces. Thus it can be drawn as a regular figure in 11-space, although then its hemi-icosahedral cells are skew; that is, each cell is not contained within a flat 3-dimensional subspace.
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Finally, a graph is Hamiltonian if there exists a cycle that passes through each of its vertices exactly once. Barnette's conjecture states that every cubic bipartite polyhedral graph is Hamiltonian. By Steinitz's theorem, a planar graph represents the edges and vertices of a convex polyhedron if and only if it is polyhedral. A three-dimensional polyhedron has a cubic graph if and only if it is a simple polyhedron.
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When determining the best suited alignments for each MSA, a trace is usually generated. A trace is a set of realized, or corresponding and aligned, vertices that has a specific weight based on the edges that are selected between corresponding vertices. When choosing traces for a set of sequences it is necessary to choose a trace with a maximum weight to get the best alignment of the sequences.
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That is, the lines connecting the vertices of the tangential triangle and the corresponding vertices of the reference triangle are concurrent. The center of perspectivity, where these three lines meet, is the symmedian point of the triangle. The tangent lines containing the sides of the tangential triangle are called the exsymmedians of the reference triangle. Any two of these are concurrent with the third symmedian of the reference triangle.
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Add a source vertex s and connect it to all the vertices in A′ and add a sink vertex t and connect all vertices inside group B′ to this vertex. The capacity of all the new edges is 1 and their costs is 0. It is proved that there is minimum weight perfect bipartite matching in G if and only if there a minimum cost flow in G′.
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For example, consider the following graphs:Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985, Chapter 5. : File:Maximum-matching-labels.svg In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cover.
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In this encoding, the notion of induced substructure is more restrictive than the notion of subgraph. For example, let G be a graph consisting of two vertices connected by an edge, and let H be the graph consisting of the same vertices but no edges. H is a subgraph of G, but not an induced substructure. The notion in graph theory that corresponds to induced substructures is that of induced subgraphs.
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There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 cells and 6 edges. Both the above great circle paths have dual great circle paths in the 600-cell. The 10 cell face to face path above maps to a 10 vertices path solely traversing along edges in the 600-cell, forming a decagon.
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The Petersen graph as a Moore graph. Any breadth first search tree has d(d-1)i vertices in its ith level. Let G be any graph with maximum degree d and diameter k, and consider the tree formed by breadth first search starting from any vertex v. This tree has 1 vertex at level 0 (v itself), and at most d vertices at level 1 (the neighbors of v).
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Sum coloring of a tree. The sum of the labels is 11, smaller than could be achieved using only two labels. In graph theory, a sum coloring of a graph is a labeling of its vertices by positive integers, with no two adjacent vertices having equal labels, that minimizes the sum of the labels. The minimum sum that can be achieved is called the chromatic sum of the graph.
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By Veblen's theorem,. every Eulerian subgraph of a given graph can be decomposed into simple cycles, subgraphs in which all vertices have degree zero or two and in which the degree-two vertices form a connected set. Therefore, it is always possible to find a basis in which the basis elements are themselves all simple cycles. Such a basis is called a cycle basis of the given graph.
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The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 8-orthoplex. The trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).
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The smallest possible number of vertices for a non-hamiltonian polyhedral graph is 11. Therefore, the Goldner–Harary graph is a minimal example of graphs of this type. However, the Herschel graph, another non-Hamiltonian polyhedron with 11 vertices, has fewer edges. As a non-Hamiltonian maximal planar graph, the Goldner–Harary graph provides an example of a planar graph with book thickness greater than two.. See in particular Figure 9.
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A semi-symmetric graph must be bipartite, and its automorphism group must act transitively on each of the two vertex sets of the bipartition (in fact, regularity is not required for this property to hold). For instance, in the diagram of the Folkman graph shown here, green vertices can not be mapped to red ones by any automorphism, but every two vertices of the same color are symmetric with each other.
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With a process graph, the vertices of the graph are of two types, operation (O) and material (M). These vertex types form two disjunctive sets. The edges of the graph link the O and M vertices. An edge from an operation vertex (O) connects to a material vertex (M) if M is the output of O, such as a 'document' (material) that is output by a 'write- up' (operation).
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One way of sharing vertex data between triangles is the triangle strip. With strips of triangles each triangle shares one complete edge with one neighbour and another with the next. Another way is the triangle fan which is a set of connected triangles sharing one central vertex. With these methods vertices are dealt with efficiently resulting in the need to only process N+2 vertices in order to draw N triangles.
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It is represented by the vertices and edges of the quarter cubic honeycomb, filling space by regular tetrahedra and truncated tetrahedra. It contains four sets of parallel planes of points and lines, each plane being a two dimensional kagome lattice. A second expression in three dimensions has parallel layers of two dimensional lattices and is called an orthorhombic- kagome lattice. The trihexagonal prismatic honeycomb represents its edges and vertices.
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For instance, the condition that a graph does not have any isolated vertices may be expressed by the sentence :\forall u\exists v(u\sim v) where the \sim symbol indicates the undirected adjacency relation between two vertices. This sentence can be interpreted as meaning that for every vertex u there is another vertex v that is adjacent to u., Section 1.2, "What Is a First Order Theory?", pp. 15–17.
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Recently, Hyperbolic Geometric Graphs have been suggested as yet another way of constructing scale-free networks. Some networks with a power-law degree distribution (and specific other types of structure) can be highly resistant to the random deletion of vertices—i.e., the vast majority of vertices remain connected together in a giant component. Such networks can also be quite sensitive to targeted attacks aimed at fracturing the network quickly.
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The smallest 8-crossing cubic graphs include the Nauru graph and the McGee graph or (3,7)-cage graph, with 24 vertices. The smallest 11-crossing cubic graphs include the Coxeter graph with 28 vertices. In 2009, Pegg and Exoo conjectured that the smallest cubic graph with crossing number 13 is the Tutte–Coxeter graph and the smallest cubic graph with crossing number 170 is the Tutte 12-cage.
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By Cauchy's rigidity theorem, a flexible polyhedron must be non-convex, but there exist other flexible polyhedra without self-crossings. However, avoiding self- crossings requires more vertices (at least nine) than the six vertices of the Bricard octahedra.. In his publication describing these octahedra, Bricard completely classified the flexible octahedra. His work in this area was later the subject of lectures by Henri Lebesgue at the Collège de France.
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Symmetries of a regular icositetragon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center. The regular icositetragon has Dih24 symmetry, order 48. There are 7 subgroup dihedral symmetries: (Dih12, Dih6, Dih3), and (Dih8, Dih4, Dih2 Dih1), and 8 cyclic group symmetries: (Z24, Z12, Z6, Z3), and (Z8, Z4, Z2, Z1).
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For all primes up to , only in two cases: and , where is the number of vertices in the cycle of 1 in the doubling diagram modulo . Here the doubling diagram represents the directed graph with the non-negative integers less than m as vertices and with directed edges going from each vertex x to vertex 2x reduced modulo m. It was shown, that for all odd prime numbers either or .
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Buckminsterfullerene is a truncated icosahedron with 60 vertices and 32 faces (20 hexagons and 12 pentagons where no pentagons share a vertex) with a carbon atom at the vertices of each polygon and a bond along each polygon edge. The van der Waals diameter of a molecule is about 1.01 nanometers (nm). The nucleus to nucleus diameter of a molecule is about 0.71 nm. The molecule has two bond lengths.
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Otherwise, it is called a weakly connected graph if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a disconnected graph. A k-vertex-connected graph or k-edge-connected graph is a graph in which no set of vertices (respectively, edges) exists that, when removed, disconnects the graph. A k-vertex-connected graph is often called simply a k-connected graph.
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Toida's conjecture refines Ádám's conjecture by considering only a special class of circulant graphs, in which all of the differences between adjacent graph vertices are relatively prime to the number of vertices. According to this refined conjecture, these special circulant graphs should have the property that all of their symmetries come from symmetries of the underlying additive group of numbers modulo . It was proven by two groups in 2001 and 2002.
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The pentagonal bifrustum is the dual polyhedron of a Johnson solid, the elongated pentagonal bipyramid. This polyhedron can be constructed by taking a pentagonal bipyramid and truncating the polar axis vertices. In Conway's notation for polyhedra, it can be represented as the polyhedron "t5dP5", meaning the truncation of the degree- five vertices of the dual of a pentagonal prism.Conway Notation for Polyhedra, George W. Hart, accessed 2014-12-20.
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A bond joining two triangular faces breaks to form a square, and then a new bond forms across opposite vertices of the square. Wandering atoms was a puzzle solved by Lipscomb in one of his few papers with no co-authors. Compounds of boron and hydrogen tend to form closed cage structures. Sometimes the atoms at the vertices of these cages move substantial distances with respect to each other.
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An outerplanar graph is biconnected if and only if the outer face of the graph forms a simple cycle without repeated vertices. An outerplanar graph is Hamiltonian if and only if it is biconnected; in this case, the outer face forms the unique Hamiltonian cycle.; . More generally, the size of the longest cycle in an outerplanar graph is the same as the number of vertices in its largest biconnected component.
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Assigning distinct colors to distinct vertices always yields a proper coloring, so : 1 \le \chi(G) \le n. The only graphs that can be 1-colored are edgeless graphs. A complete graph K_n of n vertices requires \chi(K_n)=n colors. In an optimal coloring there must be at least one of the graph’s m edges between every pair of color classes, so : \chi(G)(\chi(G)-1) \le 2m.
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Symmetric TSP with four cities TSP can be modelled as an undirected weighted graph, such that cities are the graph's vertices, paths are the graph's edges, and a path's distance is the edge's weight. It is a minimization problem starting and finishing at a specified vertex after having visited each other vertex exactly once. Often, the model is a complete graph (i.e., each pair of vertices is connected by an edge).
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The quiver itself can be considered a category, where the vertices are objects and paths are morphisms. Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces. Morphisms of representations of Q are precisely natural transformations between the corresponding functors. For a finite quiver Γ (a quiver with finitely many vertices and edges), let KΓ be its path algebra.
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A Hamiltonian path or traceable path is a path that visits each vertex of the graph exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once.
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The distance is the length of a shortest path connecting the vertices. Unless lengths of edges are explicitly provided, the length of a path is the number of edges in it. The distance matrix resembles a high power of the adjacency matrix, but instead of telling only whether or not two vertices are connected (i.e., the connection matrix, which contains boolean values), it gives the exact distance between them.
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In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, n-1.
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Also, the vertex list contains a list of faces connected to each vertex. Unlike VV meshes, both faces and vertices are explicit, so locating neighboring faces and vertices is constant time. However, the edges are implicit, so a search is still needed to find all the faces surrounding a given face. Other dynamic operations, such as splitting or merging a face, are also difficult with face-vertex meshes.
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Two slightly different cabinet shapes were released - one with straighter edges and vertices in profile and one with more rounded edges akin to its stablemate, Tehkan World Cup.
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The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.
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Its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either.
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3D model of a (uniform) heptagonal prism. In geometry, the heptagonal prism is a prism with heptagonal base. This polyhedron has 9 faces, 21 edges, and 14 vertices..
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An optimization (not necessary for the analysis) is to remove from G each edge that is found to connect two vertices in the same component as each other.
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The Rado graph may also be formed by a construction resembling that for Paley graphs, taking as the vertices of a graph all the prime numbers that are congruent to 1 modulo 4, and connecting two vertices by an edge whenever one of the two numbers is a quadratic residue modulo the other. By quadratic reciprocity and the restriction of the vertices to primes congruent to 1 mod 4, this is a symmetric relation, so it defines an undirected graph, which turns out to be isomorphic to the Rado graph. Another construction of the Rado graph shows that it is an infinite circulant graph, with the integers as its vertices and with an edge between each two integers whose distance (the absolute value of their difference) belongs to a particular set S. To construct the Rado graph in this way, S may be chosen randomly, or by choosing the indicator function of S to be the concatenation of all finite binary sequences., Section 1.2.
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The algorithm can be understood as identifying the strong component of a vertex u as the set of vertices which are reachable from u both by backwards and forwards traversal. Writing F(u) for the set of vertices reachable from u by forward traversal, B(u) for the set of vertices reachable from u by backwards traversal, and P(u) for the set of vertices which appear strictly before u on the list L after phase 2 of the algorithm, the strong component containing a vertex u appointed as root is : B(u) \cap F(u) = B(u) \setminus (B(u) \setminus F(u)) = B(u) \setminus P(u) . Set intersection is computationally costly, but it is logically equivalent to a double set difference, and since B(u) \setminus F(u) \subseteq P(u) it becomes sufficient to test whether a newly encountered element of B(u) has already been assigned to a component or not.
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A universal graph for a family of graphs can also refer to a member of a sequence of finite graphs that contains all graphs in ; for instance, every finite tree is a subgraph of a sufficiently large hypercube graph so a hypercube can be said to be a universal graph for trees. However it is not the smallest such graph: it is known that there is a universal graph for -vertex trees, with only vertices and edges, and that this is optimal.. A construction based on the planar separator theorem can be used to show that -vertex planar graphs have universal graphs with edges, and that bounded-degree planar graphs have universal graphs with edges. It is also possible to construct universal graphs for planar graphs that have vertices. Sumner's conjecture states that tournaments are universal for polytrees, in the sense that every tournament with vertices contains every polytree with vertices as a subgraph.
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The Hadwiger–Nelson problem asks how many colors are needed to color the points of the Euclidean plane in such a way that each pair of points at unit distance from each other are assigned different colors. That is, it asks for the chromatic number of the infinite graph whose vertices are all the points in the plane and whose edges are all pairs of points at unit distance. The Moser spindle requires four colors in any graph coloring: in any three-coloring of one of the two rhombi from which it is formed, the two acute-angled vertices of the rhombi would necessarily have the same color as each other. But if the shared vertex of the two rhombi has the same color as the two opposite acute-angled vertices, then these two vertices have the same color as each other, violating the requirement that the edge connecting them have differently-colored endpoints.
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However, an algorithm that is given as input an implicit graph of this type must operate on it only through the implicit adjacency test, without reference to how the test is implemented. A graph property is the question of whether a graph belongs to a given family of graphs; the answer must remain invariant under any relabeling of the vertices. In this context, the question to be determined is how many pairs of vertices must be tested for adjacency, in the worst case, before the property of interest can be determined to be true or false for a given implicit graph. Rivest and Vuillemin proved that any deterministic algorithm for any nontrivial graph property must test a quadratic number of pairs of vertices.. The full Aanderaa–Karp–Rosenberg conjecture is that any deterministic algorithm for a monotonic graph property (one that remains true if more edges are added to a graph with the property) must in some cases test every possible pair of vertices.
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Brouwer has confirmed by computation that the conjecture is valid for all graphs with at most 10 vertices. It is also known that the conjecture is valid for any number of vertices if t = 1, 2, n − 1, and n. For certain types of graphs, Brouwer's conjecture is known to be valid for all t and for any number of vertices. In particular, it is known that is valid for trees, and for unicyclic and bicyclic graphs. It was also proved that Brouwer’s conjecture holds for two large families of graphs; the first family of graphs is obtained from a clique by identifying each of its vertices to a vertex of an arbitrary c-cyclic graph, and the second family is composed of the graphs in which the removal of the edges of the maximal complete bipartite subgraph gives a graph each of whose non-trivial components is a c-cyclic graph.
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The bipartite double cover of G has two vertices ui and wi for each vertex vi of G. Two vertices ui and wj are connected by an edge in the double cover if and only if vi and vj are connected by an edge in G. For instance, below is an illustration of a bipartite double cover of a non-bipartite graph G. In the illustration, each vertex in the tensor product is shown using a color from the first term of the product (G) and a shape from the second term of the product (K2); therefore, the vertices ui in the double cover are shown as circles while the vertices wi are shown as squares. :Image:Covering-graph-2.svg The bipartite double cover may also be constructed using adjacency matrices (as described below) or as the derived graph of a voltage graph in which each edge of G is labeled by the nonzero element of the two-element group.
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A graph with odd-crossing number 13 and pair-crossing number 15. In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs (connected pieces of Jordan curves) joining the corresponding pairs of points. The points representing the vertices of a graph and the arcs representing its edges are called the vertices and the edges of the topological graph. It is usually assumed that any two edges of a topological graph cross a finite number of times, no edge passes through a vertex different from its endpoints, and no two edges touch each other (without crossing).
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The constraint hypergraph of a constraint satisfaction problem is a hypergraph in which the vertices correspond to the variables, and the hyperedges correspond to the constraints. A set of vertices forms a hyperedge if the corresponding variables are those occurring in some constraint. A simple way to represent the constraint hypergraph is by using a classical graph with the following properties: # Vertices correspond either to variables or to constraints, # an edge can only connect a variable-vertex to a constraint-vertex, and # there is an edge between a variable-vertex and a constraint-vertex if and only if the corresponding variable occurs in the corresponding constraint. Properties 1 and 2 define a bipartite graph.
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One way to construct a fixed-parameter tractable algorithm for the nonblocker problem is to use kernelization, an algorithmic design principle in which a polynomial-time algorithm is used to reduce a larger problem instance to an equivalent instance whose size is bounded by a function of the parameter. For the nonblocker problem, an input to the problem consists of a graph G and a parameter k, and the goal is to determine whether G has a nonblocker with k or more vertices. This problem has an easy kernelization that reduces it to an equivalent problem with at most 2k vertices. First, remove all isolated vertices from G, as they cannot be part of any nonblocker.
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NSPE NSPE allows the user to hand edit the polygons on NURBS surfaces. This includes being able to drag the vertices anywhere along the NURBS surface as well as join the vertices together, break the vertices apart and color them. NSPE has a significant advantage over simply converting a NURBS surface to a polygon mesh for editing because NSPE lets the user be able continue to modify the NURBS surface for the hand edited polygon structure. Because NSPE ensures that when a polygon's vertex is dragged it will always be on the NURBS surface, NSPE greatly helps the user to avoid unintentionally changing the shape of the model when optimizing for real time animation.
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An undirected graph G is Hamiltonian if it contains a cycle that touches each of its vertices exactly once. It is 2-vertex-connected if it does not have an articulation vertex, a vertex whose deletion would leave the remaining graph disconnected. Not every 2-vertex- connected graph is Hamiltonian; counterexamples include the Petersen graph and the complete bipartite graph K2,3. The square of G is a graph G2 that has the same vertex set as G, and in which two vertices are adjacent if and only if they have distance at most two in G. Fleischner's theorem states that the square of a finite 2-vertex-connected graph with at least three vertices must always be Hamiltonian.
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In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, appeared already in the 13th century, in the work of Ramon Llull.. Such a drawing is sometimes referred to as a mystic rose..
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The set of homomorphisms from to can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore, the subgraphs of a multigraph are in bijection with the graph homomorphisms from to the multigraph definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of as the multigraph , called the power object of . What is special about a multigraph as an algebra is that its operations are unary.
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The total chromatic number χ″(G) of a graph G is the fewest colors needed in any total coloring of G. The total graph T = T(G) of a graph G is a graph such that (i) the vertex set of T corresponds to the vertices and edges of G and (ii) two vertices are adjacent in T if and only if their corresponding elements are either adjacent or incident in G. Then total coloring of G becomes a (proper) vertex coloring of T(G). A total coloring is a partitioning of the vertices and edges of the graph into total independent sets. Some inequalities for χ″(G): # χ″(G) ≥ Δ(G) + 1. # χ″(G) ≤ Δ(G) + 1026.
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The Moser spindle embedded as a unit distance graph in the plane, together with a seven-coloring of the plane. As a unit distance graph, the Moser spindle is formed by two rhombi with 60 and 120 degree angles, so that the sides and short diagonals of the rhombi form equilateral triangles. The two rhombi are placed in the plane, sharing one of their acute-angled vertices, in such a way that the remaining two acute-angled vertices are a unit distance apart from each other. The eleven edges of the graph are the eight rhombus sides, the two short diagonals of the rhombi, and the edge between the unit-distance pair of acute-angled vertices.
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The Schönhardt polyhedron is combinatorially equivalent to the regular octahedron: its vertices, edges, and faces can be placed in one-to-one correspondence with the features of a regular octahedron. However, unlike the regular octahedron, three of its edges have concave dihedral angles, and these three edges form a perfect matching of the graph of the octahedron; this fact is sufficient to show that it cannot be triangulated. The six vertices of the Schönhardt polyhedron can be used to form fifteen unordered pairs of vertices. Twelve of these fifteen pairs form edges of the polyhedron: there are six edges in the two equilateral triangle faces, and six edges connecting the two triangles.
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The Dryad runtime parallelizes the dataflow graph by distributing the computational vertices across various execution engines (which can be multiple processor cores on the same computer or different physical computers connected by a network, as in a cluster). Scheduling of the computational vertices on the available hardware is handled by the Dryad runtime, without any explicit intervention by the developer of the application or administrator of the network. The flow of data between one computational vertex to another is implemented by using communication "channels" between the vertices, which in physical implementation is realized by TCP/IP streams, shared memory or temporary files. A stream is used at runtime to transport a finite number of structured Items.
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The edges of the truncated tetrahedron form a vertex- transitive graph (also a Cayley graph) which is not symmetric. Finite vertex- transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) are also vertex- transitive, as are the vertices and edges of the Archimedean solids (though only two of these are symmetric). Potočnik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices.. Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs.
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Another equivalent formulation of the union-closed sets conjecture uses graph theory. In an undirected graph, an independent set is a set of vertices no two of which are adjacent to each other; an independent set is maximal if it is not a subset of a larger independent set. In any graph, the "heavy" vertices that appear in more than half of the maximal independent sets must themselves form an independent set, so there always exists at least one non- heavy vertex, a vertex that appears in at most half of the maximal independent sets. The graph formulation of the union-closed sets conjecture states that every finite non-empty graph contains two adjacent non-heavy vertices.
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With stroke-based fonts, the same stroke paths can be filled with different stroke profiles resulting in different visual shapes without the need to specify the vertex positions of each outline, as is the case with outline fonts. A glyph's outline is defined by the vertices of individual stroke paths, and the corresponding stroke profiles. The stroke paths are a kind of topological skeleton of the glyph. The advantages of stroke-based fonts over outline fonts include reducing number of vertices needed to define a glyph, allowing the same vertices to be used to generate a font with a different weight, glyph width, or serifs using different stroke rules, and the associated size savings.
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Parallel to these edge-linked octahedral chains are vertex-linked mixed chains of alternating octahedra and tetrahedra. The tetrahedra have a phosphorus ion P in the middle, and oxygen ions O at each of the four vertices, and the octahedra have an aluminium ion Al in the middle surrounded by six oxygen ions O, as in the octahedral chains. At each linked vertex one O is shared between a tetrahedron and an octahedron, and each tetrahedron and octahedron must have two linked vertices to form the mixed chain. Each octahedral chain is flanked by two mixed chains, one on either side, linked through the vertices of the chains, making an infinite triple chain.
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Pocchiola and Vegter (1996a,b,c) originally defined a pseudotriangle to be a simply-connected region of the plane bounded by three smooth convex curves that are tangent at their endpoints. However, subsequent work has settled on a broader definition that applies more generally to polygons as well as to regions bounded by smooth curves, and that allows nonzero angles at the three vertices. In this broader definition, a pseudotriangle is a simply-connected region of the plane, having three convex vertices. The three boundary curves connecting these three vertices must be convex, in the sense that any line segment connecting two points on the same boundary curve must lie entirely outside or on the boundary of the pseudotriangle.
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Ends of graphs were defined by in terms of equivalence classes of infinite paths.However, as point out, ends of graphs were already considered by . A ' in an infinite graph is a semi-infinite simple path; that is, it is an infinite sequence of vertices v0, v1, v2, ... in which each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays r0 and r1 are equivalent if there is another ray r2 (not necessarily different from either of the first two rays) that contains infinitely many of the vertices in each of r0 and r1.
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Each 4-connected (in the above sense) simple cubic graph on vertices defines a class of quantum mechanical j symbols. Roughly speaking, each vertex represents a 3-jm symbol, the graph is converted to a digraph by assigning signs to the angular momentum quantum numbers , the vertices are labelled with a handedness representing the order of the three (of the three edges) in the 3jm symbol, and the graph represents a sum over the product of all these numbers assigned to the vertices. There are 1 (6j), 1 (9j), 2 (12j), 5 (15j), 18 (18j), 84 (21j), 607 (24j), 6100 (27j), 78824 (30j), 1195280 (33j), 20297600 (36j), 376940415 (39j) etc. of these .
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The data space for the Marching Squares algorithm is 2D, because the vertices assigned a data value are connected to their neighbors in a 2D topological grid, but the spatial coordinates assigned to the vertices can be in 2D, 3D or higher dimensions. For example, a triangular mesh may represent a 2D data surface embedded in 3D space, where spatial positions of the vertices and interpolated points along a contour will all have 3 coordinates. Note that the case of squares is ambiguous again, because a quadrilateral embedded in 3-dimensional space is not necessarily planar, so there is a choice of geometrical interpolation scheme to draw the banded surfaces in 3D.
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An outerplanar graph is an undirected graph that can be drawn in the plane without crossings in such a way that all of the vertices belong to the unbounded face of the drawing. That is, no vertex is totally surrounded by edges. Alternatively, a graph G is outerplanar if the graph formed from G by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.. A maximal outerplanar graph is an outerplanar graph that cannot have any additional edges added to it while preserving outerplanarity. Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every bounded face of a maximal outerplanar graph is a triangle.
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A penny graph with 11 vertices and 19 edges that requires four colors in any graph coloring A four-coloring of the graph above. In geometric graph theory, a penny graph is a contact graph of unit circles. That is, it is an undirected graph whose vertices can be represented by unit circles, with no two of these circles crossing each other, and with two adjacent vertices if and only if they are represented by tangent circles.. See especially p. 176. More simply, they are the graphs formed by arranging pennies in a non-overlapping way on a flat surface, making a vertex for each penny, and making an edge for each two pennies that touch.
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One way of constructing the Perles configuration is to start with a regular pentagon and its five diagonals, which form the sides of a smaller regular pentagon within the initial one. The nine points of the configuration consist of four out of the five vertices of each pentagon and the shared center of the two pentagons; the two missing pentagon vertices are chosen to be collinear with the center. The nine lines of the configuration consist of the five lines that are diagonals of the outer pentagon and sides of the inner pentagon, and the four lines that pass through the center and through corresponding pairs of vertices from the two pentagons.
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In this case each matrix can be encoded as a directed edge of a graph with n vertices. So all matrices together give a graph on n vertices with 2n directed edges. The identity holds provided that for any two vertices A and B of the graph, the number of odd Eulerian paths from A to B is the same as the number of even ones. (Here a path is called odd or even depending on whether its edges taken in order give an odd or even permutation of the 2n edges.) Swan showed that this was the case provided the number of edges in the graph is at least 2n, thus proving the Amitsur–Levitzki theorem.
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Cartesian coordinates for the vertices of a nonconvex great rhombicuboctahedron centered at the origin with edge length 1 are all the permutations of : (±ξ, ±1, ±1), where ξ = − 1\.
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Deciding whether the number of vertices of a given polytope is bounded by some natural number k is a computationally difficult problem and complete for the complexity class PP.
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In graph theory, a fractional matching is a generalization of a matching in which, intuitively, each vertex may be broken into fractions that are matched to different neighbor vertices.
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The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.
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The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.
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The equation of a line with variable distances p, q, r from the vertices A, B, C whose opposite sides are a, b, c is :apx+bqy+crz=0.
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If a pair of vertices u and v on a graph G admit perfect state transfer at time t, then both u and v are periodic at time 2t.
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It is NP-complete, given an -vertex cubic graph G and a parameter , to determine whether G can be obtained as a quotient of a planar graph with vertices..
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That part of the four-member ring, exclusive of the shared edge, has 2 carbons. The edge itself, exclusive of the two vertices that define it, has 0 carbons.
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In the mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges. It is the 8-vertex Möbius ladder graph.
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The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n-gons at the truncated vertices, and nonregular hexagonal faces.
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In mathematics, the Wirtinger plane sextic curve, studied by Wirtinger, is a degree 6 genus 4 plane curve with double points at the 6 vertices of a complete quadrilateral.
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They then consider the vertices in order, for each vertex choosing its color to be the minimum excluded value of the set of colors already assigned to its neighbors.
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The related clique edge cover problem concerns partitioning the edges of a graph, rather than the vertices, into subgraphs induced by cliques. It is also NP-complete., Problem GT59.
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For similar reasons, the cutwidth is at most the pathwidth times the maximum degree of the vertices in a given graph., Lemma 1, p. 99; , Theorem 49, p. 24.
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If the weights are positive, then a minimum spanning tree is in fact a minimum-cost subgraph connecting all vertices, since subgraphs containing cycles necessarily have more total weight.
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Roughly speaking, a Feynman diagram is called connected if all vertices and propagator lines are linked by a sequence of vertices and propagators of the diagram itself. If one views it as an undirected graph it is connected. The remarkable relevance of such diagrams in QFTs is due to the fact that they are sufficient to determine the quantum partition function . More precisely, connected Feynman diagrams determine :i W[J]\equiv \ln Z[J].
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A citation graph having vertices representing the papers in the 1994–2000 Graph Drawing symposia and having edges representing citations between these papers was made available as part of the graph drawing contest associated with the 2001 symposium.. The largest connected component of this graph consists of 249 vertices and 642 edges; clustering analysis reveals several prominent subtopics within graph drawing that are more tightly connected, including three-dimensional graph drawing and orthogonal graph drawing..
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Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols The Schläfli symbol of a (convex) regular polygon with p edges is {p}. For example, a regular pentagon is represented by {5}. For (nonconvex) star polygons, the constructive notation {} is used, where p is the number of vertices and q - 1 is the number of vertexes skipped when drawing each edge of the star. For example, {} represents the pentagram.
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A mesh generated from an implicit surface Many meshing techniques are built on the principles of the Delaunay triangulation, together with rules for adding vertices, such as Ruppert's algorithm. A distinguishing feature is that an initial coarse mesh of the entire space is formed, then vertices and triangles are added. In contrast, advancing front algorithms start from the domain boundary, and add elements incrementally filling up the interior. Hybrid techniques do both.
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He had earlier discovered a similar polytope, the 57-cell (Coxeter 1982, 1984). By 1994 Grünbaum was considering polytopes abstractly as combinatorial sets of points or vertices, and was unconcerned whether faces were planar. As he and others refined these ideas, such sets came to be called abstract polytopes. An abstract polytope is defined as a partially ordered set (poset), whose elements are the polytope's faces (vertices, edges, faces etc.) ordered by containment.
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An output of a quantitative historical linguistic analysis is normally a tree or a network diagram. This allows summary visualisation of the output data but is not the complete result. A tree is a connected acyclic graph, consisting of a set of vertices (also known as "nodes") and a set of edges ("branches") each of which connects a pair of vertices. An internal node represents a linguistic ancestor in a phylogenic tree or network.
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The star chromatic number of Dyck graph is 4, while the chromatic number is 2. In graph-theoretic mathematics, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by .
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There are other natural generalizations of bipartite graphs. A hypergraph is called balanced if it is essentially 2-colorable, and remains essentially 2-colorable upon deleting any number of vertices (see Balanced hypergraph). The properties of bipartiteness and balance do not imply each other. Bipartiteness does not imply balance. For example, let H be the hypergraph with vertices {1,2,3,4} and edges: > { {1,2,3} , {1,2,4} , {1,3,4} } It is bipartite by the partition X={1}, Y={2,3,4}.
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Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Likewise where three such lines intersect at a point that is not a corner of any face, these points are false vertices. The images below show spheres at the true vertices, and blue rods along the true edges. For example, the small stellated dodecahedron has 12 pentagram faces with the central pentagonal part hidden inside the solid.
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Topologically, the deltoidal hexecontahedron is identical to the nonconvex rhombic hexecontahedron. The deltoidal hexecontahedron can be derived from a dodecahedron (or icosahedron) by pushing the face centers, edge centers and vertices out to different radii from the body center. The radii are chosen so that the resulting shape has planar kite faces each such that vertices go to degree-3 corners, faces to degree-five corners, and edge centers to degree-four points.
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Suppose we have a complete graph on vertices. We wish to show (for small enough values of ) that it is possible to color the edges of the graph in two colors (say red and blue) so that there is no complete subgraph on vertices which is monochromatic (every edge colored the same color). To do so, we color the graph randomly. Color each edge independently with probability of being red and of being blue.
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A spanning subgraph of a given graph G has the same set of vertices as G itself but, possibly, fewer edges. A graph G, or one of its subgraphs, is said to be Eulerian if each of its vertices has even degree (its number of incident edges). Every simple cycle in a graph is an Eulerian subgraph, but there may be others. The cycle space of a graph is the collection of its Eulerian subgraphs.
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John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular 120-gons.
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John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) He gives r128 for the full reflective symmetry, Dih64, and a1 for no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.
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In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s can reach a vertex t (and t is reachable from s) if there exists a sequence of adjacent vertices (i.e. a path) which starts with s and ends with t. In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph.
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A universal traversal sequence is a sequence of instructions comprising a graph traversal for any regular graph with a set number of vertices and for any starting vertex. A probabilistic proof was used by Aleliunas et al. to show that there exists a universal traversal sequence with number of instructions proportional to for any regular graph with n vertices. The steps specified in the sequence are relative to the current node, not absolute.
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Let and be any two of the four points given by the incenter and the three excenters of a triangle . Then and are collinear with one of the three triangle vertices. The circle with as diameter passes through the other two vertices and is centered on the circumcircle of . When one of or is the incenter, this is the trillium theorem, with line as the (internal) angle bisector of one of the triangle's angles.
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Multigraphs of both Königsberg Bridges and Five room puzzles have more than two odd vertices (in orange), thus are not Eulerian and hence the puzzles have no solutions. degree. Therefore, this is an Eulerian graph. Following the edges in alphabetical order gives an Eulerian circuit/cycle. In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices).
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This implies that a strongly connected tournament has a Hamiltonian cycle (Camion 1959). More strongly, every strongly connected tournament is vertex pancyclic: for each vertex v, and each k in the range from three to the number of vertices in the tournament, there is a cycle of length k containing v., Theorem 1. Moreover, if the tournament is 4‑connected, each pair of vertices can be connected with a Hamiltonian path (Thomassen 1980).
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3D animation is digitally modeled and manipulated by an animator. The animator usually starts by creating a 3D polygon mesh to manipulate. A mesh typically includes many vertices that are connected by edges and faces, which give the visual appearance of form to a 3D object or 3D environment. Sometimes, the mesh is given an internal digital skeletal structure called an armature that can be used to control the mesh by weighting the vertices.
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Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2. The number of vertices V is then g/p2 and the number of edges E is g/p1. The complex polygon illustrated above has eight square edges (p1=4) and sixteen vertices (p2=2).
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B.Sz. Nagy, L. Rédey: Eine Verallgemeinerung der Inhaltsformel von Heron. Publ. Math. Debrecen 1, 42–50 (1949) The signed area depends on the ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive -axis to the positive -axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative.
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In words, when , one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or ≲ is used instead of ≤. To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive.
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The Symmetries of Things, Chapter 20 r2000000 represents full symmetry and a1 labels no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. These lower symmetries allows degrees of freedom in defining irregular megagons. Only the g1000000 subgroup has no degrees of freedom but can be seen as directed edges.
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Prismatoid with parallel faces A₁ and A₃, midway cross-section A₂, and height h In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles.William F. Kern, James R Bland, Solid Mensuration with proofs, 1938, p.75 If both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a prismoid.
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Microsoft made several preview releases of this technology available as add-ons to Windows HPC Server 2008 R2. An application written for Dryad is modeled as a directed acyclic graph (DAG). The DAG defines the dataflow of the application, and the vertices of the graph defines the operations that are to be performed on the data. The "computational vertices" are written using sequential constructs, devoid of any concurrency or mutual exclusion semantics.
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Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2. The number of vertices V is then g/p2 and the number of edges E is g/p1. The complex polygon illustrated above has eight square edges (p1=4) and sixteen vertices (p2=2).
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All vertices of a finite n-dimensional isogonal figure exist on an (n-1)-sphere. The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory. The pseudorhombicuboctahedronwhich is not isogonaldemonstrates that simply asserting that "all vertices look the same" is not as restrictive as the definition used here, which involves the group of isometries preserving the polyhedron or tiling.
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The relation between the number of vertices, edges and faces of any finite polyhedron is given by Euler's polyhedron formula: : e - f - v = 2 (g -1),\, where e, f and v are the number of edges, faces and vertices, respectively, and g is the genus of the polyhedron, i.e., the number of "holes" in the surface. For example, a sphere is a surface of genus 0, while a torus is of genus 1.
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The webgraph describes the directed links between pages of the World Wide Web. A graph, in general, consists of several vertices, some pairs connected by edges. In a directed graph, edges are directed lines or arcs. The webgraph is a directed graph, whose vertices correspond to the pages of the WWW, and a directed edge connects page X to page Y if there exists a hyperlink on page X, referring to page Y.
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An abstract dependency graph is a directed graph, a graph of vertices connected by one-way edges. Most often, the vertices and edges of the graph represent the inputs and outputs of functions in or components of the system. By inspecting the created abstract dependency graph, the developer can detect syntactic anomalies (or Preece anomalies) in the system. While anomalies are not always defects, they often provide clues to finding defects in a system.
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Branko Grünbaum constructed an example of a non- polytopal simplicial sphere (that is, a simplicial sphere that is not the boundary of a polytope). Gil Kalai proved that, in fact, "most" simplicial spheres are non-polytopal. The smallest example is of dimension d = 4 and has f0 = 8 vertices. The upper bound theorem gives upper bounds for the numbers fi of i-faces of any simplicial d-sphere with f0 = n vertices.
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The oxychlorides are only very slightly soluble in water. In the system MgO – – at about 23 °C, the completely liquid region has vertices at the following triple equilibrium points (as mass fractions, not molar fractions): : S1 = 0.008 MgO + 0.170 + 0.822 (Sol::P5) : S2 = 0.010 MgO + 0.222 + 0.768 (Sol:P5:P3) : S3 = 0.012 MgO + 0.345 + 0.643 (Sol:P3:·6) The other vertices are pure water, magnesium chloride hexahydrate, and the saturated solution (0.0044 MgO + 0.9956 by mass).
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Vertex figure for the omnisnub tesseract The full snub tesseract or omnisnub tesseract, defined as an alternation of the omnitruncated tesseract, can not be made uniform, but it can be given Coxeter diagram , and symmetry [4,3,3]+, and constructed from 8 snub cubes, 16 icosahedra, 24 square antiprisms, 32 octahedra (as triangular antiprisms), and 192 tetrahedra filling the gaps at the deleted vertices. It has 272 cells, 944 faces, 864 edges, and 192 vertices.
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Jenő Egerváry (1931) considered graphs in which each edge e has a non- negative integer weight we. The weight vector is denoted by w. The w-weight of a matching is the sum of weights of the edges participating in the matching. A w-vertex-cover is a multiset of vertices ("multiset" means that each vertex may appear several times), in which each edge e is adjacent to at least we vertices.
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In 1972, McMullen proposed the following problem:D. G. Larman (1972), "On Sets Projectively Equivalent to the Vertices of a Convex Polytope", Bulletin of the London Mathematical Society 4, pp.6-12 : Determine the largest number u(d) such that for any given u(d) points in general position in affine d-space Rd there is a projective transformation mapping these points into convex position (so they form the vertices of a convex polytope).
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A polyhedron realized from a circle packing. The circles representing the vertices of the polyhedron are their horizons on the sphere, and the circles representing the faces (dual vertices) are the intersections of the sphere with the face planes. According to one variant of the circle packing theorem, for every polyhedral graph and its dual graph, there exists a system of circles in the plane or on any sphere, representing the vertices of both graphs, so that two adjacent vertices in the same graph are represented by tangent circles, a primal and dual vertex that represent a vertex and face that touch each other are represented by orthogonal circles, and all remaining pairs of circles are disjoint.. From such a representation on a sphere, one can find a polyhedral realization of the given graph as the intersection of a collection of halfspaces, one for each circle that represents a dual vertex, with the boundary of the halfspace containing the circle. Alternatively and equivalently, one can find the same polyhedron as the convex hull of a collection of points (its vertices), such that the horizon seen when viewing the sphere from any vertex equals the circle that corresponds to that vertex.
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A variation of the planar separator theorem involves edge separators, small sets of edges forming a cut between two subsets A and B of the vertices of the graph. The two sets A and B must each have size at most a constant fraction of the number n of vertices of the graph (conventionally, both sets have size at most 2n/3), and each vertex of the graph belongs to exactly one of A and B. The separator consists of the edges that have one endpoint in A and one endpoint in B. Bounds on the size of an edge separator involve the degree of the vertices as well as the number of vertices in the graph: the planar graphs in which one vertex has degree n − 1, including the wheel graphs and star graphs, have no edge separator with a sublinear number of edges, because any edge separator would have to include all the edges connecting the high degree vertex to the vertices on the other side of the cut. However, every planar graph with maximum degree Δ has an edge separator of size O(√(Δn)). proved this result for 2-connected planar graphs, and extended it to all planar graphs.
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In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph..
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In graph theory, a folded cube graph is an undirected graph formed from a hypercube graph by adding to it a perfect matching that connects opposite pairs of hypercube vertices.
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3D model of a tetradyakis hexahedron The tetradyakis hexahedron (or great disdyakis dodecahedron) is a nonconvex isohedral polyhedron. It has 48 intersecting scalene triangle faces, 72 edges, and 20 vertices.
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Stated precisely, in any graph G its maximal pseudoforests consist of every tree component of G, together with one or more disjoint 1-trees covering the remaining vertices of G.
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One of Ochiai's unknots featuring 139 vertices, for example, was originally unknotted by computer in 108 hours, but this time has been reduced in more recent research to 10 minutes.
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The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex.
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The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 7-orthoplex.
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The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.
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The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on facets of the birectified 7-orthoplex.
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The vertices of the stericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 8-orthoplex.
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The vertices of the bistericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 8-orthoplex.
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The vertices of the steritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 8-orthoplex.
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The vertices of the bisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,3). This construction is based on facets of the bisteritruncated 8-orthoplex.
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The vertices of the stericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 8-orthoplex.
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The vertices of the bistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,3). This construction is based on facets of the stericantellated 8-orthoplex.
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The vertices of the stericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,4). This construction is based on facets of the stericantitruncated 8-orthoplex.
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The vertices of the bistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,4). This construction is based on facets of the bistericantitruncated 8-orthoplex.
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The vertices of the steriruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,2,3). This construction is based on facets of the steriruncinated 8-orthoplex.
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The vertices of the steriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,4). This construction is based on facets of the steriruncitruncated 8-orthoplex.
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The vertices of the steriruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,3,4). This construction is based on facets of the steriruncicantellated 8-orthoplex.
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The vertices of the bisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,4). This construction is based on facets of the bisteriruncitruncated 8-orthoplex.
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The vertices of the steriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 8-orthoplex.
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The vertices of the bisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,5,5). This construction is based on facets of the bisteriruncicantitruncated 8-orthoplex.
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The vertices of the pentellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,1,2). This construction is based on facets of the pentellated 8-orthoplex.
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The vertices of the pentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,3). This construction is based on facets of the pentitruncated 8-orthoplex.
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The vertices of the penticantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,2,3). This construction is based on facets of the penticantellated 8-orthoplex.
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The vertices of the penticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 8-orthoplex.
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The vertices of the pentiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,2,3). This construction is based on facets of the pentiruncinated 8-orthoplex.
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The vertices of the pentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,4). This construction is based on facets of the pentiruncitruncated 8-orthoplex.
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The vertices of the pentiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,3,4). This construction is based on facets of the pentiruncicantellated 8-orthoplex.
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The vertices of the pentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,5). This construction is based on facets of the pentiruncicantitruncated 8-orthoplex.
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The vertices of the pentistericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,2,3). This construction is based on facets of the pentistericated 8-orthoplex.
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The vertices of the pentisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,4). This construction is based on facets of the pentisteritruncated 8-orthoplex.
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The vertices of the pentistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,3,4). This construction is based on facets of the pentistericantellated 8-orthoplex.
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The vertices of the pentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,5). This construction is based on facets of the pentistericantitruncated 8-orthoplex.
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The vertices of the pentisteriruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,3,3,4). This construction is based on facets of the pentisteriruncinated 8-orthoplex.
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The vertices of the pentisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,3,4,5). This construction is based on facets of the pentisteriruncitruncated 8-orthoplex.
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The vertices of the pentisteriruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,4,5). This construction is based on facets of the pentisteriruncicantellated 8-orthoplex.
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The vertices of the pentisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,5,6). This construction is based on facets of the pentisteriruncicantitruncated 8-orthoplex.
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The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .
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The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .
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The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .
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The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .
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The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, .
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The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, .
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The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, .
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The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, .
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The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, .
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The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, .
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The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, .
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The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .
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The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .
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The vertices of the hexipenticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, .
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The vertices of the hexisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, .
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The vertices of the hexisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, .
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The vertices of the hexipentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,5,6). This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex, .
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The vertices of the hexipentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,3,4,5,6). This construction is based on facets of the hexipentistericantitruncated 8-orthoplex, .
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The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.
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The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.
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The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.
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The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.
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The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.
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The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.
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The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex.
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The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex.
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The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex.
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The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex.
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The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex.
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The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex.
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The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex.
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The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex.
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The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 7-orthoplex.
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The vertices of the runcicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 7-orthoplex.
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The vertices of the biruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 7-orthoplex.
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The vertices of the stericated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,1,2). This construction is based on facets of the stericated 7-orthoplex.
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The vertices of the stericanttruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the stericantitruncated 7-orthoplex.
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The vertices of the steriruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,2,3,3). This construction is based on facets of the steriruncinated 7-orthoplex.
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The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.
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The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.
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The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.
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The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.
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The vertices of the runcinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,1,2). This construction is based on facets of the runcinated 7-orthoplex.
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The vertices of the biruncinted 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 7-orthoplex.
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The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.
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The vertices of the biruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 7-orthoplex.
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The vertices of the steritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 7-orthoplex.
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The vertices of the stericantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 7-orthoplex.
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The vertices of the steriruncittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncitruncated 7-orthoplex.
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The vertices of the steriruncitcantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncicantellated 7-orthoplex.
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The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 7-orthoplex.
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But it also runs slowly, taking time O(m2n) on a network of n vertices and m edges, making it impractical for networks of more than a few thousand nodes.
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Petersen graph. Graphs are fundamental objects in combinatorics. Considerations of graph theory range from enumeration (e.g., the number of graphs on n vertices with k edges) to existing structures (e.g.
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The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex.
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The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 8-orthoplex.
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The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 8-orthoplex.
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The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 8-orthoplex.
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The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 8-orthoplex.
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According to the classical four-vertex theorem, every simple closed planar smooth curve must have at least four vertices., Theorem 9.3.9, p. 570; , Section 9.3, "The Four Vertex Theorem", pp.
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Models start with a single node v_0 and have k self-loops. v_t denotes a vertex added in the t^{th} step, and n denotes the total number of vertices.
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However, Joseph Horton provided a counterexample on 96 vertices, the Horton graph.Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 240, 1976.
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A tree T with 2n vertices, is bivariegated if and only if the independence number of T is n, or, equivalently, if and only if it has a perfect matching.
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The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.
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The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the penticantellated 7-orthoplex.
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The vertices of the penticantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 7-orthoplex.
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The vertices of the pentiruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the pentiruncitruncated 7-orthoplex.
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The vertices of the pentiruncicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,2,3,3,4). This construction is based on facets of the pentiruncicantellated 7-orthoplex.
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The vertices of the pentiruncicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,2,3,4,5). This construction is based on facets of the pentiruncicantitruncated 7-orthoplex.
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The vertices of the pentisteritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,2,2,3,4). This construction is based on facets of the pentisteritruncated 7-orthoplex.
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The vertices of the pentistericantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,2,3,4,5). This construction is based on facets of the pentistericantitruncated 7-orthoplex.
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The urogenital triangle is the area bound by a triangle with one vertex at the pubic symphysis and the two other vertices at the ischial tuberosities of the pelvic bone.
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If the centre of the hexagon is 0 and the vertices in order are a, b, c, −a, −b and −c, then Λ is the Abelian group with generators and .
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More broadly, a deltoid can refer to any closed figure with three vertices connected by curves that are concave to the exterior, making the interior points a non-convex set.
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3D model of a truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
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A triangular prism has 6 vertices, 9 edges, bounded by 2 triangular and 3 quadrilateral faces. The advantage with this type of layer is that it resolves boundary layer efficiently.
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Each triangle can be extraverted in three different ways; the 18 Morley triangles and 27 extravert pairs of triangles form the 18 vertices and 27 edges of the Pappus graph..
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Every ideal polyhedron with n vertices has a surface that can be subdivided into 2n-4 ideal triangles,See, e.g., p. 272 of . each with area \pi., Proposition 2.4.12, p. 83.
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The Cartesian coordinates for the 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations: : (±1,±1,±3,±3,±5) with an odd number of plus signs.
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The Cartesian coordinates for the 320 vertices of a steriruncic 5-cube centered at the origin are coordinate permutations: : (±1,±1,±1,±3,±5) with an odd number of plus signs.
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The Cartesian coordinates for the 960 vertices of a steriruncicantic 5-cube centered at the origin are coordinate permutations: : (±1,±1,±3,±5,±7) with an odd number of plus signs.
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The larger neighbors of vertex 1 are all vertices with numbers congruent to 2 or 3 modulo 4, because those are exactly the numbers with a nonzero bit at index 1.; .
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A Static Mesh contains information about its shape (vertices, edges and sides), a reference to the textures to be used, and optionally a collision model (see the simple collision section below).
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The goal for the player is to eliminate all of the crossings and construct a straight-line embedding of the graph by moving the vertices one by one into better positions.
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Each of the eight 3-digit sequences (corresponding to the eight vertices) appears exactly twice, and each of the sixteen 4-digit sequences (corresponding to the 16 edges) appears exactly once.
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Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract: : (±1,±1,±1,±1,±1,±1) with an odd number of plus signs.
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More generally, a polygon in which all vertices are concyclic is called a cyclic polygon. A polygon is cyclic if and only if the perpendicular bisectors of its edges are concurrent..
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If such a solution exists, then x1, ..., xn are the tangent lengths of the polygon (the lengths from the vertices to the points where the incircle is tangent to the sides).
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In the mathematical field of graph theory, the Livingstone graph is a distance-transitive graph with 266 vertices and 1463 edges. It is the largest distance-transitive graph with degree 11.
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The distance matrix of can be computed from as above, however, calculated by the usual matrix multiplication only encodes the number of paths between any two vertices of length at most .
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The Cartesian coordinates for the 960 vertices of a runcic 5-cubes centered at the origin are coordinate permutations: : (±1,±1,±1,±3,±3) with an odd number of plus signs.
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The Cartesian coordinates for the 480 vertices of a runcicantic 5-cube centered at the origin are coordinate permutations: : (±1,±1,±3,±5,±5) with an odd number of plus signs.
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The Cartesian coordinates for the vertices of a runcic 6-cube centered at the origin are coordinate permutations: : (±1,±1,±1,±3,±3,±3) with an odd number of plus signs.
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The Cartesian coordinates for the vertices of a runcicantic 6-cube centered at the origin are coordinate permutations: : (±1,±1,±3,±5,±5,±5) with an odd number of plus signs.
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The Cartesian coordinates for the vertices of a cantellated demihepteract centered at the origin are coordinate permutations: : (±1,±1,±1,±3,±3,±3,±3) with an odd number of plus signs.
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The reachability relationship in any directed acyclic graph can be formalized as a partial order on the vertices of the DAG. In this partial order, two vertices and are ordered as exactly when there exists a directed path from to in the DAG; that is, when is reachable from .. However, different DAGs may give rise to the same reachability relation and the same partial order.. For example, the DAG with two edges and has the same reachability relation as the graph with three edges , , and . Both of these DAGS produce the same partial order, in which the vertices are ordered as . If is a DAG, its transitive closure is the graph with the most edges that represents the same reachability relation.
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Because G is triangular, the degree of each vertex in a configuration is known, and all edges internal to the configuration are known, the number of vertices in G adjacent to a given configuration is fixed, and they are joined in a cycle. These vertices form the ring of the configuration; a configuration with k vertices in its ring is a k-ring configuration, and the configuration together with its ring is called the ringed configuration. As in the simple cases above, one may enumerate all distinct four-colorings of the ring; any coloring that can be extended without modification to a coloring of the configuration is called initially good. For example, the single-vertex configuration above with 3 or less neighbors were initially good.
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From the model theoretic point of view, the Rado graph is an example of a saturated model. This is just a logical formulation of the property that the Rado graph contains all finite graphs as induced subgraphs. In this context, a type is a set of variables together with a collection of constraints on the values of some or all of the predicates determined by those variables; a complete type is a type that constrains all of the predicates determined by its variables. In the theory of graphs, the variables represent vertices and the predicates are the adjacencies between vertices, so a complete type specifies whether an edge is present or absent between every pair of vertices represented by the given variables.
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The cube and the octahedron, two examples for which the bound of the conjecture is tight In two dimensions, the simplest centrally symmetric convex polygons are the parallelograms, which have four vertices, four edges, and one polygon; . A cube is centrally symmetric, and has 8 vertices, 12 edges, 6 square sides, and 1 solid; . Another three-dimensional convex polyhedron, the regular octahedron, is also centrally symmetric, and has 6 vertices, 12 edges, 8 triangular sides, and 1 solid; . In higher dimensions, the hypercube [0,1]d has exactly 3d faces, each of which can be determined by specifying, for each of the d coordinate axes, whether the face projects onto that axis onto the point 0, the point 1, or the interval [0,1].
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There exist inputs to the travelling salesman problem that cause the Christofides algorithm to find a solution whose approximation ratio is arbitrarily close to 3/2. One such class of inputs are formed by a path of vertices, with the path edges having weight , together with a set of edges connecting vertices two steps apart in the path with weight for a number chosen close to zero but positive. All remaining edges of the complete graph have distances given by the shortest paths in this subgraph. Then the minimum spanning tree will be given by the path, of length , and the only two odd vertices will be the path endpoints, whose perfect matching consists of a single edge with weight approximately .
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Any counting formula involving vertices and faces that is valid for all planar graphs may be transformed by planar duality into an equivalent formula in which the roles of the vertices and faces have been swapped. Euler's formula, which is self-dual, is one example. Another given by Harary involves the handshaking lemma, according to which the sum of the degrees of the vertices of any graph equals twice the number of edges. In its dual form, this lemma states that in a plane graph, the sum of the numbers of sides of the faces of the graph equals twice the number of edges.. The medial graph of a plane graph is isomorphic to the medial graph of its dual.
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Unlike tree traversal, graph traversal may require that some vertices be visited more than once, since it is not necessarily known before transitioning to a vertex that it has already been explored. As graphs become more dense, this redundancy becomes more prevalent, causing computation time to increase; as graphs become more sparse, the opposite holds true. Thus, it is usually necessary to remember which vertices have already been explored by the algorithm, so that vertices are revisited as infrequently as possible (or in the worst case, to prevent the traversal from continuing indefinitely). This may be accomplished by associating each vertex of the graph with a "color" or "visitation" state during the traversal, which is then checked and updated as the algorithm visits each vertex.
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It is also possible to define geodesics on some surfaces that are not smooth everywhere, such as convex polyhedra. The surface of a convex polyhedron has a metric that is locally Euclidean except at the vertices of the polyhedron, and a curve that avoids the vertices is a geodesic if it follows straight line segments within each face of the polyhedron and stays straight across each polyhedron edge that it crosses. Although some polyhedra have simple closed geodesics (for instance, the regular tetrahedron and disphenoids have infinitely many closed geodesics, all simple).. others do not. In particular, a simple closed geodesic of a convex polyhedron would necessarily bisect the total angular defect of the vertices, and almost all polyhedra do not have such bisectors.
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For instance in a simple graph, we can define the overlap between two k-cliques to be the number of vertices common to both k-cliques. The Clique Percolation Method is then equivalent to thresholding this clique graph, dropping all edges of weight less than (k-1), with the remaining connected components forming the communities of cliques found in CPM. For k=2 the cliques are the edges of the original graph and the clique graph in this case is the line graph of the original network. In practice, using the number of common vertices as a measure of the strength of clique overlap may give poor results as large cliques in the original graph, those with many more than k vertices, will dominate the clique graph.
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To construct these graphs, Henson orders the vertices of the Rado graph into a sequence with the property that, for every finite set of vertices, there are infinitely many vertices having as their set of earlier neighbors. (The existence of such a sequence uniquely defines the Rado graph.) He then defines to be the induced subgraph of the Rado graph formed by removing the final vertex (in the sequence ordering) of every -clique of the Rado graph. With this construction, each graph is an induced subgraph of , and the union of this chain of induced subgraphs is the Rado graph itself. Because each graph omits at least one vertex from each -clique of the Rado graph, there can be no -clique in .
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In the mathematical field of graph theory, the Chang graphs are a set of three 12-regular undirected graphs, each with 28 vertices and 168 edges. They are strongly regular, with the same parameters and spectra as the line graph L(K8) of the complete graph K8. Each of these three graphs may be obtained by graph switching from L(K8). That is, a subset S of the vertices of L(K8) is chosen, each edge that connects a vertex in S with a vertex not in S is deleted from L(K8), and an edge is added for each pair of vertices (with again one in S and one not in S) that were not already connected by an edge.
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In the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using formulas of mathematical logic. There are several variations in the types of logical operation that can be used in these formulas. The first order logic of graphs concerns formulas in which the variables and predicates concern individual vertices and edges of a graph, while monadic second order graph logic allows quantification over sets of vertices or edges. Logics based on least fixed point operators allow more general predicates over tuples of vertices, but these predicates can only be constructed through fixed-point operators, restricting their power to an intermediate level between first order and monadic second order.
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Oettingen and Riemann's Tonnetz thus extended on infinitely in every direction without actually repeating any pitches. The appeal of the Tonnetz to 19th- century German theorists was that it allows spatial representations of tonal distance and tonal relationships. For example, looking at the dark blue A minor triad in the graphic at the beginning of the article, its parallel major triad (A-C#-E) is the triangle right below, sharing the vertices A and E. The relative major of A minor, C major (C-E-G) is the upper-right adjacent triangle, sharing the C and the E vertices. The dominant triad of A minor, E major (E-G#-B) is diagonally across the E vertex, and shares no other vertices.
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It can be realized as a projective polyhedron (a tessellation of the real projective plane by three quadrilaterals), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts. It has three square faces, six edges, and four vertices. It has an unexpected property that every face is in contact with every other face on two edges, and every face contains all the vertices, which gives an example of an abstract polytope whose faces are not determined by their vertex sets. From the point of view of graph theory the skeleton is a tetrahedral graph, an embedding of K4 (the complete graph with four vertices) on a projective plane.
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A domatic partitionIn graph theory, a domatic partition of a graph G = (V,E) is a partition of V into disjoint sets V_1, V_2,...,V_K such that each Vi is a dominating set for G. The figure on the right shows a domatic partition of a graph; here the dominating set V_1 consists of the yellow vertices, V_2 consists of the green vertices, and V_3 consists of the blue vertices. The domatic number is the maximum size of a domatic partition, that is, the maximum number of disjoint dominating sets. The graph in the figure has domatic number 3. It is easy to see that the domatic number is at least 3 because we have presented a domatic partition of size 3.
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In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v1 and v2 of G, there is some automorphism :f\colon G \to G\ such that :f(v_1) = v_2.\ In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices.. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph).
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A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipathGraph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991, p.205) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.
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221 polytope. This symmetric projection contains 2 rings of 12 vertices, and 3 vertices coinciding at the center. The intersection graph of the 27 lines on a cubic surface is a locally linear graph that is the complement of the Schläfli graph. That is, two vertices are adjacent in the Schläfli graph if and only if the corresponding pair of lines are skew.. The Schläfli graph may also be constructed from the system of eight-dimensional vectors :(1, 0, 0, 0, 0, 0, 1, 0), :(1, 0, 0, 0, 0, 0, 0, 1), and :(−1/2, −1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2), and the 24 other vectors obtained by permuting the first six coordinates of these three vectors.
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A graph is said to be -factor-critical if every subset of vertices has a perfect matching. Under this definition, a hypomatchable graph is 1-factor-critical.. Even more generally, a graph is -factor-critical if every subset of vertices has an -factor, that is, it is the vertex set of an -regular subgraph of the given graph. A critical graph (without qualification) is usually assumed to mean a graph for which removing each of its vertices reduces the number of colors it needs in a graph coloring. The concept of criticality has been used much more generally in graph theory to refer to graphs for which removing each possible vertex changes or does not change some relevant property of the graph.
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If S is a maximal independent set in some graph, it is a maximal clique or maximal complete subgraph in the complementary graph. A maximal clique is a set of vertices that induces a complete subgraph, and that is not a subset of the vertices of any larger complete subgraph. That is, it is a set S such that every pair of vertices in S is connected by an edge and every vertex not in S is missing an edge to at least one vertex in S. A graph may have many maximal cliques, of varying sizes; finding the largest of these is the maximum clique problem. Some authors include maximality as part of the definition of a clique, and refer to maximal cliques simply as cliques.
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Any 3-CNF formula may be converted into a graph by constructing a separate gadget for each of its variables and clauses and connecting them as shown.This reduction is described in . In any 3-coloring of the resulting graph, one may designate the three colors as being true, false, or ground, where false and ground are the colors given to the false and ground vertices (necessarily different, as these vertices are made adjacent by the construction) and true is the remaining color not used by either of these vertices. Within a variable gadget, only two colorings are possible: the vertex labeled with the variable must be colored either true or false, and the vertex labeled with the variable's negation must correspondingly be colored either false or true.
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A data structure constructed from the minimum spanning tree allows the minimax distance between any pair of vertices to be queried in constant time per query, using lowest common ancestor queries in a Cartesian tree. The root of the Cartesian tree represents the heaviest minimum spanning tree edge, and the children of the root are Cartesian trees recursively constructed from the subtrees of the minimum spanning tree formed by removing the heaviest edge. The leaves of the Cartesian tree represent the vertices of the input graph, and the minimax distance between two vertices equals the weight of the Cartesian tree node that is their lowest common ancestor. Once the minimum spanning tree edges have been sorted, this Cartesian tree can be constructed in linear time.
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Like the hypercube graph, the vertices of the Fibonacci cube of order n may be labeled with bitstrings of length n, in such a way that two vertices are adjacent whenever their labels differ in a single bit. However, in a Fibonacci cube, the only labels that are allowed are bitstrings with no two consecutive 1 bits. There are Fn + 2 labels possible, where Fn denotes the nth Fibonacci number, and therefore there are Fn + 2 vertices in the Fibonacci cube of order n. Fibonacci cubes (drawn in red) as subgraphs of hypercubes The nodes of such a network may be assigned consecutive integers from 0 to Fn + 2 − 1; the bitstrings corresponding to these numbers are given by their Zeckendorf representations.
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We record here definitions from . Let be the Coxeter complex associated to a group W generated by a set of reflections S. The vertices of are the elements of W, and the chambers of the complex are the cosets of S in W. The vertices of each chamber can be colored in a one-to-one manner by the elements of S so that no adjacent vertices of the complex receive the same color. This coloring, although essentially canonical, is not quite unique. The coloring of a given chamber is not uniquely determined by its realization as a coset of S. But once the coloring of a single chamber has been fixed, the rest of the Coxeter complex is uniquely colorable.
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To a graph G there corresponds a signed complete graph Σ on the same vertex set, whose edges are signed negative if in G and positive if not in G. Conversely, G is the subgraph of Σ that consists of all vertices and all negative edges. The two-graph of G can also be defined as the set of triples of vertices that support a negative triangle (a triangle with an odd number of negative edges) in Σ. Two signed complete graphs yield the same two-graph if and only if they are equivalent under switching. Switching of G and of Σ are related: switching the same vertices in both yields a graph H and its corresponding signed complete graph.
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A path decomposition can be described as a sequence of graphs Gi that are glued together by identifying pairs of vertices from consecutive graphs in the sequence, such that the result of performing all of these gluings is G. The graphs Gi may be taken as the induced subgraphs of the sets Xi in the first definition of path decompositions, with two vertices in successive induced subgraphs being glued together when they are induced by the same vertex in G, and in the other direction one may recover the sets Xi as the vertex sets of the graphs Gi. The width of the path decomposition is then one less than the maximum number of vertices in one of the graphs Gi.
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The circumconic and the circumcircle share a fourth point, X(110) of the reference triangle. Finally there are two interesting and documented circumcubics that pass through the six vertices of the reference triangle and its Johnson triangle as well as the circumcenter, the orthocenter and the nine-point center. The first is known as the first Musselman cubic – K026. This cubic also passes through the six vertices of the medial triangle and the medial triangle of the Johnson triangle.
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To prove that transitive reduction is as hard as transitive closure, Aho et al. construct from a given directed acyclic graph G another graph H, in which each vertex of G is replaced by a path of three vertices, and each edge of G corresponds to an edge in H connecting the corresponding middle vertices of these paths. In addition, in the graph H, Aho et al. add an edge from every path start to every path end.
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The points dualize to lines and the convex hull of the points dualizes to the upper and lower envelope of the set of lines. The vertices of the upper convex hull dualize to segments on the upper envelope. The vertices of the lower convex hull dualize to segments on the lower envelope. The range of slopes of the supporting lines of a point on the hull dualize to the x-interval of segment that point dualizes to.
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A curved triangle patch. Normals at vertices are used to recursively subdivide the triangle into four sub-triangles In order to improve geometric fidelity, the format allows curving the triangle patches. By default, all triangles are assumed to be flat and all triangle edges are assumed to be straight lines connecting their two vertices. However, curved triangles and curved edges can optionally be specified in order to reduce the number of mesh elements required to describe a curved surface.
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The points dualize to lines and the convex hull of the points dualizes to the upper and lower envelope of the set of lines. The vertices of the upper convex hull dualize to segments on the upper envelope. The vertices of the lower convex hull dualize to segments on the lower envelope. The range of slopes of the supporting lines of a point on the hull dualize to the x-interval of segment that point dualizes to.
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Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are concyclic.
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Example of mixed graph Mixed graph coloring can be thought of as a labeling or an assignment of different colors (where is a positive integer) to the vertices of a mixed graph. Different colors must be assigned to vertices that are connected by an edge. The colors may be represented by the numbers from to , and for a directed arc, the tail of the arc must be colored by a smaller number than the head of the arc.
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In the balanced assignment problem, both parts of the bipartite graph have the same number of vertices, denoted by n. One of the first polynomial-time algorithms for balanced assignment was the Hungarian algorithm. It is a global algorithm – it is based on improving a matching along augmenting paths (alternating paths between unmatched vertices). Its run-time complexity, when using Fibonacci heaps, is O(mn + n^2\log n), where m is a number of edges.
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