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 -free induced subgraphs of G in which every two nonadjacent vertices are connected by at least p internally disjoint paths.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1520,
author = {Terry A. Mckee},
title = {Clique graph representations of ptolemaic graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {30},
year = {2010},
pages = {651-661},
zbl = {1217.05168},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1520}
}
Terry A. Mckee. Clique graph representations of ptolemaic graphs. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 651-661. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1520/
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