Discussiones Mathematicae Graph Theory 30(4) (2010) 651-661
doi: 10.7151/dmgt.1520

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CLIQUE GRAPH REPRESENTATIONS OF PTOLEMAIC GRAPHS

Terry A. McKee

Department of Mathematics and Statistics
Wright State University
Dayton, Ohio 45435, USA

Abstract

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 Pn+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.

2010 Mathematics Subject Classification: 05C62, 05C75.

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Received 17 April 2009
Revised 23 February 2010
Accepted 2 March 2010