Clique graph representations of ptolemaic graphs

A graph is ptolemaic if and only if it is both chordal and distance-hereditary. Thus, a ptolemaic graph G has two kinds of intersection graph representations: one from being chordal, and the other from being distance-hereditary. The first of these, called a clique tree representation, is easily generated from the clique graph of G (the intersection graph of the maximal complete subgraphs of G). The second intersection graph representation can also be generated from the clique graph, as a very special case of the main result: The maximal Pₙ-free connected induced subgraphs of the p-clique graph of a ptolemaic graph G correspond in a natural way to the maximal ${\displaystyle P_{n+1}}$-free induced subgraphs of G in which every two nonadjacent vertices are connected by at least p internally disjoint paths.