The Q App tracks both sleep and activity only in a circle graph that you either complete or don't, based on your set goals.
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In addition to seeing details like gender and age, graphed out as a circle graph and bar chart, respectively, the follower analytics' section also lets you track how many new followers you've gained on an hourly and daily basis.
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Every distance-hereditary graph is a circle graph, as is every permutation graph and every indifference graph. Every outerplanar graph is also a circle graph.; .
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The chords forming the 220-vertex 5-chromatic triangle-free circle graph of , drawn as an arrangement of lines in the hyperbolic plane. The chromatic number of a circle graph is the minimum number of colors that can be used to color its chords so that no two crossing chords have the same color. Since it is possible to form circle graphs in which arbitrarily large sets of chords all cross each other, the chromatic number of a circle graph may be arbitrarily large, and determining the chromatic number of a circle graph is NP-complete. It remains NP-complete to test whether a circle graph can be colored by four colors.
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A circle graph, the intersection graph of chords of a circle. For book embeddings with a fixed vertex order, finding the book thickness is equivalent to coloring a derived circle graph. Finding the book thickness of a graph is NP-hard. This follows from the fact that finding Hamiltonian cycles in maximal planar graphs is NP-complete.
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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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On the left a set of polygons inscribed in a circle; on the right the relative Polygon-circle graph (intersection graph of the polygons). On the bottom the alternating sequence of polygons around the circle. In the mathematical discipline of graph theory, a polygon-circle graph is an intersection graph of a set of convex polygons all of whose vertices lie on a common circle. These graphs have also been called spider graphs.
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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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Therefore circle graphs capture various aspects of this routing problem. Colorings of circle graphs may also be used to find book embeddings of arbitrary graphs: if the vertices of a given graph G are arranged on a circle, with the edges of G forming chords of the circle, then the intersection graph of these chords is a circle graph and colorings of this circle graph are equivalent to book embeddings that respect the given circular layout. In this equivalence, the number of colors in the coloring corresponds to the number of pages in the book embedding.
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55, No. 289 (Mar. 1960), pp. 38–70. Playfair invented several types of diagrams: in 1786 the line, area and bar chart of economic data, and in 1801 the pie chart and circle graph, used to show part- whole relations.Michael Friendly (2008).
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Australasian J. Combin. 69 (2017), 105−118. it is shown that any 132-representable graph is necessarily a circle graph, and any tree and any cycle graph, as well as any graph on at most 5 vertices, are 132-representable. It was shown in Mandelshtam.
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Contracting an edge of a polygon- circle graph results in another polygon-circle graph. A geometric representation of the new graph may be formed by replacing the polygons corresponding to the two endpoints of the contracted edge by their convex hull. Alternatively, in the alternating sequence representing the original graph, combining the subsequences representing the endpoints of the contracted edge into a single subsequence produces an alternating sequence representation of the contracted graph. Polygon circle graphs are also closed under induced subgraph or equivalently vertex deletion operations: to delete a vertex, remove its polygon from the geometric representation, or remove its subsequence of points from the alternating sequence.
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3, p. 65. Maximal outerplanar graphs are also formed as the graphs of polygon triangulations. They are examples of 2-trees, of series-parallel graphs, and of chordal graphs. Every outerplanar graph is a circle graph, the intersection graph of a set of chords of a circle.
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M. Koebe announced a polynomial time recognition algorithm,. however his preliminary version had "serious errors". and a final version was never published. Martin Pergel later proved that the problem of recognizing these graphs is NP- complete.. It is also NP-complete to determine whether a given graph can be represented as a polygon-circle graph with at most vertices per polygon, for any .
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This class of graphs was first suggested by Michael Fellows in 1988, motivated by the fact that it is closed under edge contraction and induced subgraph operations.. A polygon-circle graph can be represented as an "alternating sequence". Such a sequence can be gained by perturbing the polygons representing the graph (if necessary) so that no two share a vertex, and then listing for each vertex (in circular order, starting at an arbitrary point) the polygon attached to that vertex.
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A graph is a circle graph if and only if it is the overlap graph of a set of intervals on a line. This is a graph in which the vertices correspond to the intervals, and two vertices are connected by an edge if the two intervals overlap, with neither containing the other. The intersection graph of a set of intervals on a line is called the interval graph. String graphs, the intersection graphs of curves in the plane, include circle graphs as a special case.
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Another type of non-Euclidean geometry is the hyperbolic plane, and arrangements of hyperbolic lines in this geometry have also been studied. Any finite set of lines in the Euclidean plane has a combinatorially equivalent arrangement in the hyperbolic plane (e.g. by enclosing the vertices of the arrangement by a large circle and interpreting the interior of the circle as a Klein model of the hyperbolic plane). However, in hyperbolic line arrangements lines may avoid crossing each other without being parallel; the intersection graph of the lines in a hyperbolic arrangement is a circle graph.
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Since circle graph coloring with four or more colors is NP-hard, and since any circle graph can be formed in this way from some book embedding problem, it follows that optimal book embedding is also NP-hard.... For a fixed vertex ordering on the spine of a two-page book drawing, it is also NP-hard to minimize the number of crossings when this number is nonzero. If the spine ordering is unknown but a partition of the edges into two pages is given, then it is possible to find a 2-page embedding (if it exists) in linear time by an algorithm based on SPQR trees... However, it is NP-complete to find a 2-page embedding when neither the spine ordering nor the edge partition is known. Finding the book crossing number of a graph is also NP-hard, because of the NP-completeness of the special case of testing whether the 2-page crossing number is zero. As a consequence of bounded expansion, the subgraph isomorphism problem, of finding whether a pattern graph of bounded size exists as a subgraph of a larger graph, can be solved in linear time when the larger graph has bounded book thickness.
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This default image occasionally changes to sport a giraffe with different accessories. Past images have included a cowboy rendition (complete with cowboy hat, sunglasses, and western style lasso necklace), a bandana replacing the cap, and a heart hidden among the giraffe's many spots during the 2009 Valentine's Day edition. The graph portion of the comic generally consists of a simple line graph depicted as an arrow indicating a positive slope or negative slope, depending on the correlation that day. However, this also occasionally changes to include a more complex line graph, a circle graph, and on one occasion to-date, a graphic unrepresentative of an actual graph altogether.
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If every edge of a given graph G is subdivided, the resulting graph is a string graph if and only if G is planar. In particular, the subdivision of the complete graph K5 shown in the illustration is not a string graph, because K5 is not planar. Every circle graph, as an intersection graph of line segments (the chords of a circle), is also a string graph. Every chordal graph may be represented as a string graph: chordal graphs are intersection graphs of subtrees of trees, and one may form a string representation of a chordal graph by forming a planar embedding of the corresponding tree and replacing each subtree by a string that traces around the subtree's edges.
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14, p.164. Every distance-hereditary graph can be represented as the intersection graph of chords on a circle, forming a circle graph. This can be seen by building up the graph by adding pendant vertices, false twins, and true twins, at each step building up a corresponding set of chords representing the graph. Adding a pendant vertex corresponds to adding a chord near the endpoints of an existing chord so that it crosses only that chord; adding false twins corresponds to replacing a chord by two parallel chords crossing the same set of other chords; and adding true twins corresponds to replacing a chord by two chords that cross each other but are nearly parallel and cross the same set of other chords.
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Thus, it is possible in polynomial time to find the maximum clique or maximum independent set in a distance- hereditary graph, or to find an optimal graph coloring of any distance- hereditary graph. present a simple direct algorithm for maximum weighted independent sets in distance-hereditary graphs, based on parsing the graph into pendant vertices and twins, correcting a previous attempt at such an algorithm by . Because distance-hereditary graphs are perfectly orderable, they can be optimally colored in linear time by using LexBFS to find a perfect ordering and then applying a greedy coloring algorithm. Because distance- hereditary graphs are circle graphs, they inherit polynomial time algorithms for circle graphs; for instance, it is possible determine in polynomial time the treewidth of any circle graph and therefore of any distance-hereditary graph.
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In the particular case when k = 3 (that is, for triangle-free circle graphs) the chromatic number is at most five, and this is tight: all triangle-free circle graphs may be colored with five colors, and there exist triangle-free circle graphs that require five colors.See for the upper bound, and for the matching lower bound. and give earlier weaker bounds on the same problem. If a circle graph has girth at least five (that is, it is triangle-free and has no four-vertex cycles) it can be colored with at most three colors.. The problem of coloring triangle-free squaregraphs is equivalent to the problem of representing squaregraphs as isometric subgraphs of Cartesian products of trees; in this correspondence, the number of colors in the coloring corresponds to the number of trees in the product representation.
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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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The problem of assigning edges to the two pages in a compatible way can be formulated as either an instance of 2-satisfiability, or as a problem of testing the bipartiteness of the circle graph whose vertices are the basepairs and whose edges describe crossings between basepairs. Alternatively and more efficiently, as show, a bi-secondary structure exists if and only if the diagram graph of the input (a graph formed by connecting the bases into a cycle in their sequence order and adding the given basepairs as edges) is a planar graph. This characterization allows bi-secondary structures to be recognized in linear time as an instance of planarity testing. used the connection between secondary structures and book embeddings as part of a proof of the NP-hardness of certain problems in RNA secondary structure comparison.. And if an RNA structure is tertiary rather than bi-secondary (that is, if it requires more than two pages in its diagram), then determining the page number is again NP-hard..
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