Geodesic
Clique graph representations of ptolemaic graphs
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 651-661

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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 P_n+1-free induced subgraphs of G in which every two nonadjacent vertices are connected by at least p internally disjoint paths.
Keywords: Ptolemaic graph, clique graph, chordal graph, clique tree, graph representation 
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McKee, Terry. Clique graph representations of ptolemaic graphs. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 651-661. http://geodesic.mathdoc.fr/item/DMGT_2010_30_4_a9/