Chordal graph

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A cycle (black) with two chords (green). As for this part, the graph is chordal. However, removing one green edge would result in a non-chordal graph. Indeed, the other green edge with three black edges would form a cycle of length four with no chords.

In the mathematical area of graph theory, a graph is chordal if each of its cycles of four or more nodes has a chord, which is an edge joining two nodes that are not adjacent in the cycle. An equivalent definition is that any induced cycles have at most three nodes. Chordal graphs are a subset of the perfect graphs. They are sometimes also called triangulated graphs.

1 Perfect elimination and efficient recognition
2 Maximal cliques and graph coloring
3 Intersection graphs of subtrees
4 References
5 External links

[edit] Perfect elimination and efficient recognition

A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex v, v and the neighbors of v that occur later than v in the order form a clique. A graph is chordal if and only if it has a perfect elimination ordering (Fulkerson and Gross 1965).

Rose et al (1976; see also Habib et al 2000) show that a perfect elimination ordering of a chordal graph may be found efficiently using an algorithm known as lexicographic breadth first search. This algorithm maintains a partition of the vertices of the graph into a sequence of sets; initially this sequence consists of a single set with all vertices. The algorithm repeatedly chooses a vertex v from the earliest set in the sequence that contains previously-unchosen vertices, and splits each set S of the sequence into two smaller subsets, the first consisting of the neighbors of v in S and the second consisting of the non-neighbors. When this splitting process has been performed for all vertices, the sequence of sets has one vertex per set, in the reverse of a perfect elimination ordering.

Since both this lexicographic breadth first search process and the process of testing whether an ordering is a perfect elimination ordering can be performed in linear time, it is possible to recognize chordal graphs in linear time.

[edit] Maximal cliques and graph coloring

Another application of perfect elimination orderings is that finding the maximum clique of a chordal graph is a polynomial-time problem, while the same problem is NP-complete on general graphs. More generally, a chordal graph can have only linearly many maximal or maximum cliques, while non-chordal graphs may have exponentially many. To list all maximal cliques of a chordal graph, simply find a perfect elimination ordering, form a clique for each vertex v together with the neighbors of v that are later than v in the perfect elimination ordering, and test whether each of the resulting cliques is maximal.

The largest maximal clique is a maximum clique, and, as chordal graphs are perfect, the size of this clique equals the chromatic number of the chordal graph.

[edit] Intersection graphs of subtrees

An alternative characterization of chordal graphs, due to Gavril (1974), involves trees and their subtrees. From a collection of subtrees of a tree, one can define an intersection graph that has one vertex per subtree and an edge connecting any two subtrees that overlap in one or more nodes of the tree. As Gavril showed, the graphs that can be formed in this way are exactly the chordal graphs.

[edit] References

Fulkerson, D. R.; Gross, O. A. (1965). "Incidence matrices and interval graphs". Pacific J. Math 15: 835–855.

Gavril, Fănică (1974). "The intersection graphs of subtrees in trees are exactly the chordal graphs". Journal of Combinatorial Theory, Series B 16: 47–56.