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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]
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.
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.
A chordal graph, a special type of perfect graph, has no holes of any size greater than three. The girth of a graph is the length of its shortest cycle; this cycle is necessarily chordless. Cages are defined as the smallest regular graphs with given combinations of degree and girth.
Graph minor [4] Bipartite graphs: Odd cycles Subgraph [5] Chordal graphs: Cycles of length 4 or more Induced subgraph [6] Perfect graphs: Cycles of odd length 5 or more or their complements: Induced subgraph [7] Line graph of graphs: 9 forbidden subgraphs: Induced subgraph [8] Graph unions of cactus graphs
It is possible to determine whether a graph is strongly chordal in polynomial time, by repeatedly searching for and removing a simple vertex.If this process eliminates all vertices in the graph, the graph must be strongly chordal; otherwise, if this process finds a subgraph without any more simple vertices, the original graph cannot be strongly chordal.
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.