The graph of the 3-3 duoprism (the line graph of ) is perfect. Here it is colored with three colors, with one of its 3-vertex maximum cliques highlighted.
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.
The perfect graphs include many important families of graphs and serve to unify results relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time, despite their greater complexity for non-perfect graphs. In addition, several important minimax theorems in combinatorics, including Dilworth's theorem and Mirsky's theorem on partially ordered sets, Kőnig's theorem on matchings, and the Erdős–Szekeres theorem on monotonic sequences, can be expressed in terms of the perfection of certain associated graphs.
The perfect graph theorem states that the complement graph of a perfect graph is also perfect. The strong perfect graph theorem characterizes the perfect graphs in terms of certain forbidden induced subgraphs, leading to a polynomial time algorithm for testing whether a graph is perfect.
In graph theory, a perfectgraph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every...
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes...
In graph theory, the perfectgraph theorem of László Lovász (1972a, 1972b) states that an undirected graph is perfect if and only if its complement graph...
results concerns perfectgraphs: every bipartite graph, the complement of every bipartite graph, the line graph of every bipartite graph, and the complement...
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges...
In graph theory, the strong perfectgraph theorem is a forbidden graph characterization of the perfectgraphs as being exactly the graphs that have neither...
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...
graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G = (V, E), a perfect matching...
mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context...
the complement of a perfectgraph is also perfect is the perfectgraph theorem of László Lovász. Cographs are defined as the graphs that can be built up...
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...
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...
intersection graph of the intervals. Interval graphs are chordal graphs and perfectgraphs. They can be recognized in linear time, and an optimal graph coloring...
In graph theory, a line perfectgraph is a graph whose line graph is a perfectgraph. Equivalently, these are the graphs in which every odd-length simple...
two-vertex complete graph, and may be decomposed into two copies of Qn – 1 connected to each other by a perfect matching. Hypercube graphs should not be confused...
Meyniel graphs are a subclass of the perfectgraphs. Every induced subgraph of a Meyniel graph is another Meyniel graph, and in every Meyniel graph the size...
subclasses of the perfectgraphs, block graphs are perfect. Every tree, cluster graph, or windmill graph is a block graph. Every block graph has boxicity at...
In graph theory, a trivially perfectgraph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals...
of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching...
Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs. An incomparability...
In graph theory, a dominating set for a graph G is a subset D of its vertices, such that any vertex of G is in D, or has a neighbor in D. The domination...
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to...
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