A Characterisation of the Minimal Triangulations of Permutation Graphs

A minimal triangulation of a graph is a chordal graph obtained from adding an inclusion-minimal set of edges to the graph. For permutation graphs, i.e., graphs that are both comparability and cocomparability graphs, it is known that minimal triangulations are interval graphs. We (negatively) answer the question whether every interval graph is a minimal triangulation of a permutation graph. We give a non-trivial characterisation of the class of interval graphs that are minimal triangulations of permutation graphs and obtain as a surprising result that only “a few” interval graphs are minimal triangulations of permutation graphs. 1 Introdu tion The study of minimal triangulations of arbitrary graphs and restri ted graph lasses has a long tradition. Given a graph, a triangulation is a hordal graph that is obtained by adding edges to the graph. Minimal triangulations are spe ial triangulations. Using a hara terisation of Rose, Tarjan, Lueker, we an say that a triangulation is alled minimal, if the deletion of ea h single added edge yields a nonhordal graph [17℄. Minimal triangulations are losely onne ted to the treewidth problem, sin e the treewidth of a graph is the smallest lique number minus 1 among its minimal triangulations [16℄. Many NP-hard problems be ome tra table on graphs of bounded treewidth, so that omputing the treewidth is of highly pra ti al interest, whi h motivates the study of minimal triangulations. But also another resear h bran h motivates the study of minimal triangulations: results show that graphs and their minimal triangulations share stru tural properties. An easy result is that every graph has a minimal triangulation with the same independent-set number. Another, stronger, result is that every minimal separator of a minimal triangulation is a minimal separator also of the base graph [12℄. In this ontext also ts the fa t that minimal triangulations an be used to hara terise graphs and graph lasses. An early result shows that minimal triangulations of ographs are trivially perfe t graphs [3℄. Sin e trivially perfe t graphs are interval graphs, this also shows that treewidth and pathwidth are equal for ographs. On the other hand, ographs are exa tly the P4free graphs, and trivially perfe t graphs are exa tly the P4-free hordal graphs, from whi h follows that every minimal triangulation of a ograph is a ograph. This result was extended by Parra and S he er to the following: for k 5, a graph is Pk-free if and only if every of its minimal triangulations is Pk-free [15℄. The ograph result was generalised also in another dire tion, in parti ular to permutation graphs: Theorem 1 [2℄ Minimal triangulations of permutation graphs are interval graphs. Later, it was shown that this result even holds for o omparability graphs [11℄ and AT-free graphs [14℄. By the following hara terisation, the lass of AT-free graphs is the largest lass of graphs ontaining only graphs whose minimal triangulations are interval graphs: a graph is AT-free if and only if it has only minimal triangulations that are interval graphs [14℄, [15℄. So, for some graph lasses, minimal triangulations re e t stru tural properties of the base graphs: minimal triangulations of Pk-free graphs for k 5 are Pk-free, minimal 1