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Chordal graphs are precisely the graphs that are both odd-hole-free and even-hole-free (see holes in graph theory). Every chordal graph is a strangulated graph , a graph in which every peripheral cycle is a triangle, because peripheral cycles are a special case of induced cycles.
The chordal graphs are the graphs formed by a construction of this type in which, at the time a vertex is added, its neighbors form a clique. Chordal graphs may also be characterized as the graphs that have no holes (even or odd). [35] They include as special cases the forests, the interval graphs, [36] and the maximal outerplanar graphs. [37]
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A perfect graph is an undirected graph with the property that, in every one of its induced subgraphs, the size of the largest clique equals the minimum number of colors in a coloring of the subgraph. Perfect graphs include many important graphs classes including bipartite graphs, chordal graphs, and comparability graphs.
Cycle graph, a graph that consists of a single cycle; Chordal graph, a graph in which every induced cycle is a triangle; Directed acyclic graph, a directed graph with no directed cycles; Forest, a cycle-free graph; Line perfect graph, a graph in which every odd cycle is a triangle; Perfect graph, a graph with no induced cycles or their ...
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.
A triangle-free graph is a graph with no induced cycle of length three. The cographs are exactly the graphs with no induced path of length three. The chordal graphs are the graphs with no induced cycle of length four or more. The even-hole-free graphs are the graphs containing no induced cycles with an even number of vertices.
Every cycle of length at least 6 has a chord connecting two vertices that are a distance > 1 apart from each other in the cycle.. In the mathematical area of graph theory, a chordal bipartite graph is a bipartite graph B = (X,Y,E) in which every cycle of length at least 6 in B has a chord, i.e., an edge that connects two vertices that are a distance > 1 apart from each other in the cycle.