In graph theory, a circle graph is the intersection graph of a chord diagram. That is, it is an undirected graph whose vertices can be associated with a finite system of chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
After earlier polynomial time algorithms, [1] Gioan et al. (2013) presented an algorithm for recognizing circle graphs in near-linear time. Their method is slower than linear by a factor of the inverse Ackermann function, and is based on lexicographic breadth-first search. The running time comes from a method for maintaining the split decomposition of a graph incrementally, as vertices are added, used as a subroutine in the algorithm. [2]
A number of other problems that are NP-complete on general graphs have polynomial time algorithms when restricted to circle graphs. For instance, Kloks (1996) showed that the treewidth of a circle graph can be determined, and an optimal tree decomposition constructed, in O(n3) 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(n3) time. [3] Tiskin (2010) has shown that a maximum clique of a circle graph can be found in O(n log2n) time, while 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.
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. [4]
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. [5] It remains NP-complete to test whether a circle graph can be colored by four colors. [6] Unger (1992) claimed that finding a coloring with three colors may be done in polynomial time but his writeup of this result omits many details. [7]
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 colors. One way of stating this is that the circle graphs are -bounded. [8] 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. [9] 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. [10] 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. [11]
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. [12] Among the goals of wire routing step is to ensure that different nets stay electrically disconnected, and their potential intersecting parts must be laid out in different conducting layers. 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. [6]
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.
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. [13]
The circle graphs are generalized by the polygon-circle graphs, intersection graphs of polygons all inscribed in the same circle. [14]
In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing.
In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals.
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.
In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart. A graph is perfect when these numbers are equal, and remain equal after the deletion of arbitrary subsets of vertices.
In graph theory, an acyclic coloring is a (proper) vertex coloring in which every 2-chromatic subgraph is acyclic. The acyclic chromatic numberA(G) of a graph G is the fewest colors needed in any acyclic coloring of G.
In graph theory, a subcoloring is an assignment of colors to a graph's vertices such that each color class induces a vertex disjoint union of cliques. That is, each color class should form a cluster graph.
In graph theory, a Meyniel graph is a graph in which every odd cycle of length five or more has at least two chords. The chords may be uncrossed or they may cross each other, as long as there are at least two of them.
In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. The answer is unknown, but has been narrowed down to one of the numbers 5, 6 or 7. The correct value may depend on the choice of axioms for set theory.
In graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph, and the pathwidth of G is a number that measures how much the path was thickened to form G. More formally, a path-decomposition is a sequence of subsets of vertices of G such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets, and the pathwidth is one less than the size of the largest set in such a decomposition. Pathwidth is also known as interval thickness, vertex separation number, or node searching number.
In graph theory, the Grundy number or Grundy chromatic number of an undirected graph is the maximum number of colors that can be used by a greedy coloring strategy that considers the vertices of the graph in sequence and assigns each vertex its first available color, using a vertex ordering chosen to use as many colors as possible. Grundy numbers are named after P. M. Grundy, who studied an analogous concept for directed graphs in 1939. The undirected version was introduced by Christen & Selkow (1979).
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.
In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph.
In graph theory, a branch of discrete mathematics, a distance-hereditary graph is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph.
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.
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.
In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other. Indifference graphs are also the intersection graphs of sets of unit intervals, or of properly nested intervals. Based on these two types of interval representations, these graphs are also called unit interval graphs or proper interval graphs; they form a subclass of the interval graphs.
In graph theory, the act of coloring generally implies the assignment of labels to vertices, edges or faces in a graph. The incidence coloring is a special graph labeling where each incidence of an edge with a vertex is assigned a color under certain constraints.
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 and these are the only edges connecting any two vertices which are endpoints of the matching edges.
In graph theory, a subfield of mathematics, a well-colored graph is an undirected graph for which greedy coloring uses the same number of colors regardless of the order in which colors are chosen for its vertices. That is, for these graphs, the chromatic number and Grundy number are equal.
