On the Laminar Structure of Ptolemaic and Distance Hereditary Graphs

Ptolemaic graphs are graphs that satisfy ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs. The graph class can also be seen as a natural generalization of block graphs (and hence trees). In this paper, a new characterization of ptolemaic graphs is presented. It is a canonical tree representation based on a laminar structure of cliques. The tree representation is constructed in linear time from a perfect elimination ordering obtained by the lexicographic breadth first search. Hence recognition and graph isomorphism for ptolemaic graphs can be solved in linear time. The tree representation also gives a simple intersection model for ptolemaic graphs. The results are also extended to distance hereditary graphs.