Characterizations and algorithmic applications of chordal graph embeddings

We introduce the separator graph for a given graph G and show a 1-1 correspondence between its maximal cliques and the minimal triangulations (i.e., subset of or equal to-minimal chordal embeddings) of G. This approach can be used for characterizations of graph classes by properties of their minimal separators. In particular, we show that a graph is AT-free if and only if every minimal triangulation is an interval graph, that a graph is claw-free AT-free if and only if every minimal triangulation is a proper interval graph, and that a graph is a cograph if and only if every minimal triangulation is a trivially perfect graph. These results have algorithmic consequences for several graph parameters that are related to triangulation problems. In this context, we also show how the vertex ranking problem can be formulated as a triangulation problem into trivially perfect graphs. As consequences for the claw-free AT-free graphs we obtain that the bandwidth equals the treewidth and pathwidth, and that the proper interval completion number equals the chordal completion number and interval completion number. This directly implies that computing the bandwidth or interval completion number is NP-hard even for co-bipartite graphs and, on the other side, that there are efficient algorithms for these problems on many other claw-free subclasses of co-comparability graphs.