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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).
In the mathematical area of graph theory, an undirected graph G is strongly chordal if it is a chordal graph and every cycle of even length (≥ 6) in G has an odd chord, i.e., an edge that connects two vertices that are an odd distance (>1) apart from each other in the cycle.
graph minors, smaller graphs obtained from subgraphs by arbitrary edge contractions. The set of structures that are forbidden from belonging to a given graph family can also be called an obstruction set for that family. Forbidden graph characterizations may be used in algorithms for testing whether
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 ...
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]
A minimum chordal completion is a chordal completion with as few edges as possible. A different type of chordal completion, one that minimizes the size of the maximum clique in the resulting chordal graph, can be used to define the treewidth of G. Chordal completions can also be used to characterize several other graph classes including AT-free ...
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.
The Goldner–Harary graph is a planar graph: it can be drawn in the plane with none of its edges crossing. When drawn on a plane, all its faces are triangular, making it a maximal planar graph. As with every maximal planar graph, it is also 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph.