Chordal completion

In graph theory, a branch of mathematics, a chordal completion of a given undirected graph G is a chordal graph, on the same vertex set, that has G as a subgraph. A minimal chordal completion is a chordal completion such that any graph formed by removing an edge would no longer be a chordal completion. 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 graphs, claw-free AT-free graphs, and cographs. The minimum chordal completion was one of twelve computational problems whose complexity was listed as open in the 1979 book Computers and Intractability. Applications of chordal completion include modeling the problem of minimizing fill-in when performing Gaussian elimination on sparse symmetric matrices, and reconstructing phylogenetic trees. Chordal completions of a graph are sometimes called triangulations, but this term is ambiguous even in the context of graph theory, as it can also refer to maximal planar graphs. A graph G is an AT-free graph if and only if all of its minimal chordal completions are interval graphs. G is a claw-free AT-free graph if and only if all of its minimal chordal completions are proper interval graphs. And G is a cograph if and only if all of its minimal chordal completions are trivially perfect graphs. A graph G has treewidth at most k if and only if G has at least one chordal completion whose maximum clique size is at most k + 1. It has pathwidth at most k if and only if G has at least one chordal completion that is an interval graph with maximum clique size at most k + 1. It has bandwidth at most k if and only if G has at least one chordal completion that is a proper interval graph with maximum clique size at most k + 1.