Clique graph representations of ptolemaic graphs

Terry A. McKee
2010 Discussiones Mathematicae Graph Theory  
A graph is ptolemaic if and only if it is both chordal and distancehereditary. 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
more » ... cial case of the main result: The maximal P n -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.
doi:10.7151/dmgt.1520 fatcat:laqtj6bp3rcpziv2zmw3dixhr4