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| [[Image:Circle graph.svg|thumb|175px|A circle with five chords and the corresponding circle graph.]]
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| In [[graph theory]], a '''circle graph''' is the [[intersection graph]] of a set of [[Chord (geometry)|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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| ==Algorithmic complexity==
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| {{harvtxt|Spinrad|1994}} gives an O(''n''<sup>2</sup>)-time algorithm that tests whether a given ''n''-vertex undirected graph is a circle graph and, if it is, constructs a set of chords that represents it.
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| A number of other problems that are [[NP-complete]] on general graphs have polynomial time algorithms when restricted to circle graphs. For instance, {{harvtxt|Kloks|1996}} showed that the [[treewidth]] of a circle graph can be determined, and an optimal tree decomposition constructed, in O(''n''<sup>3</sup>) time. Additionally, a minimum fill-in (that is, a [[chordal graph]] with as few edges as possible that contains the given circle graph as a subgraph) may be found in O(''n''<sup>3</sup>) time.<ref>{{harvtxt|Kloks|Kratsch|Wong|1998}}.</ref>
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| {{harvtxt|Tiskin|2010}} has shown that a [[maximum clique]] of a circle graph can be found in O(''n''log<sup>2</sup> ''n'') time, while
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| {{harvtxt|Nash|Gregg|2010}} have shown that a [[maximum independent set]] of an unweighted circle graph can be found in O(''n''min{''d'', ''α''}) time, where ''d'' is a parameter of the graph known as its density, and ''α'' is the independence number of the circle graph.
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| However, there are also problems that remain NP-complete when restricted to circle graphs. These include the [[minimum dominating set]], minimum connected dominating set, and minimum total dominating set problems.<ref>{{harvtxt|Keil|1993}}</ref>
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| ==Chromatic number==
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| [[File:Ageev 5X circle graph.svg|thumb|left|300px|The chords forming the 220-vertex 5-chromatic triangle-free circle graph of {{harvtxt|Ageev|1996}}, drawn as an [[arrangement of lines]] in the
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| [[Hyperbolic space|hyperbolic plane]].]]
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| 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.<ref>{{harvtxt|Garey|Johnson|Miller|Papadimitriou|1980}}; {{harvtxt|Unger|1988}}.</ref> However, several authors have investigated problems of coloring restricted subclasses of circle graphs with few colors. In particular, for circle graphs in which no sets of ''k'' or more chords all cross each other, it is possible to color the graph with as few as <math>21\cdot 2^k-24k-24</math> colors.<ref>{{harvtxt|Černý|2007}}. For earlier bounds see {{harvtxt|Gyárfás|1985}}, {{harvtxt|Kostochka|1988}}, and {{harvtxt|Kostochka|Kratochvíl|1997}}.</ref> In the particular case when ''k'' = 3 (that is, for [[triangle-free graph|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.<ref>See {{harvtxt|Kostochka|1988}} for the upper bound, and {{harvtxt|Ageev|1996}} for the matching lower bound. {{harvtxt|Karapetyan|1984}} and {{harvtxt|Gyárfás|Lehel|1985}} give earlier weaker bounds on the same problem.</ref> If a circle graph has [[girth (graph theory)|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.<ref>{{harvtxt|Ageev|1999}}.</ref>
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| ==Applications==
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| Circle graphs arise in [[VLSI]] [[physical design]] as an abstract representation for a special case for [[wire routing]], known as "two-terminal [[switchbox routing]]". In this case the [[routing area]] is a rectangle, all nets are two-terminal, and the terminals are placed on the perimeter of the rectangle. It is easily seen that the intersection graph of these nets is a circle graph. <ref>Naveed Sherwani, "Algorithms for VLSI Physical Design Automation"</ref> Among the goals of wire routing step is to ensure that different nets stay electrically disconnected, and their potential intersecting parts must be [[integrated circuit layout|laid out]] in different conducting layers. Therefore circle graphs capture various aspects of this routing problem.
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| Colorings of circle graphs may also be used to find [[book embedding]]s 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.<ref>{{harvtxt|Unger|1988}}.</ref>
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| ==Related graph classes==
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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.
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| The [[intersection graph]] of a set of intervals on a line is called the [[interval graph]].
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| [[String graph]]s, the [[intersection graph]]s of curves in the plane, include circle graphs as a special case.
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| Every [[distance-hereditary graph]] is a circle graph, as is every [[permutation graph]]. Every [[outerplanar graph]] is also a circle graph.<ref>{{harvtxt|Wessel|Pöschel|1985}}; {{harvtxt|Unger|1988}}.</ref>
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| | title = A triangle-free circle graph with chromatic number 5
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| | volume = 152
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| | year = 1996
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| | doi = 10.1016/0012-365X(95)00349-2}}.
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| | title = Every circle graph of girth at least 5 is 3-colourable
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| | year = 1999
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| | doi = 10.1016/S0012-365X(98)00192-7}}.
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| | doi = 10.1016/j.endm.2007.07.072
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| | journal = Electronic Notes in Discrete Mathematics
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| | pages = 357–361
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| | title = Coloring circle graphs
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| | last3 = Miller | first3 = G. L. | author3-link = Gary Miller (professor)
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| | title = On perfect arc and chord intersection graphs
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| | year = 1984}}. As cited by {{harvtxt|Ageev|1996}}.
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| *{{citation
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| | pages = 51–63
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| | title = The complexity of domination problems in circle graphs
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| | volume = 42
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| | year = 1993}}.
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| *{{citation
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| | year = 1996
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| | title = Recognition of circle graphs
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| | publisher = Springer-Verlag
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| | editor-last = Sachs | editor-first = Horst | editor-link = Horst Sachs
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| | publisher = B.G. Teubner
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| | year = 1985}}. As cited by {{harvtxt|Unger|1988}}.
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| *{{citation
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| | last = Tiskin | first = Alexander
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| | pages = 1287–1296
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| | contribution = Fast distance multiplication of unit-Monge matrices.
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| | title = Proceedings of ACM-SIAM SODA 2010
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| | year = 2010}}.
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| *{{citation
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| | last1 = Nash | first1 = Nicholas
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| | last2 = Gregg | first2 = David
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| | issue = 16
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| | pages = 630–634
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| | title = An output sensitive algorithm for computing a maximum independent set of a circle graph
| |
| | volume = 116
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| | year = 2010}}.
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| == External links ==
| |
| | |
| *{{citation
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| | publisher = [http://www.graphclasses.org/index.html Information System on Graph Class Inclusions]
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| | title = Circle graph
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| | url = http://www.graphclasses.org/classes/gc_132.html}}
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| [[Category:Circles]]
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| [[Category:Intersection classes of graphs]]
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| [[Category:Geometric graphs]]
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