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The even-hole-free graphs are the graphs containing no induced cycles with an even number of vertices. The trivially perfect graphs are the graphs that have neither an induced path of length three nor an induced cycle of length four. By the strong perfect graph theorem, the perfect graphs are the graphs with no odd hole and no odd antihole.
While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a specific vertex. [ 3 ] It is unknown whether graph coloring and the maximum independent set problem can be solved in polynomial time on even-hole-free graphs, or whether they are NP-complete.
In algebraic topology and graph theory, graph homology describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1 ...
A similar problem is finding induced subgraphs in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it. Finding maximal induced subgraphs of a certain kind is also often NP-complete. For example:
Chordal graphs are a subclass of the well known perfect graphs. Other superclasses of chordal graphs include weakly chordal graphs, cop-win graphs, odd-hole-free graphs, even-hole-free graphs, and Meyniel graphs. Chordal graphs are precisely the graphs that are both odd-hole-free and even-hole-free (see holes in graph theory).
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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 ...
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