Two characterisations of the minimal triangulations of permutation graphs

A minimal triangulation of a graph is a chordal supergraph with an inclusion-minimal edge set. Minimal triangulations are obtained from adding edges only to minimal separators, completing minimal separators into cliques. Permutation graphs are the comparability graphs whose complements are also comparability graphs. Permutation graphs can be characterised as the intersection graphs of specially arranged line segments in the plane, which is called a permutation diagram. The minimal triangulations of permutation graphs are known to be interval graphs, and they can be obtained from permutation diagrams by applying a geometric operation, that corresponds to the completion of separators into cliques. We precisely specify this geometric completion process to obtain minimal triangulations, and we completely characterise those interval graphs that are minimal triangulations of permutation graphs.