Laminar structure of ptolemaic graphs and its applications

Authors:
Ryuhei Uehara;Yushi Uno
Affiliations:
School of Information Science, Japan Advanced Institute of Science and Technology (JAIST), Ishikawa, Japan;Department of Mathematics and Information Sciences, Graduate School of Science, Osaka Prefecture University, Sakai, Japan
Venue:
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Year:
2005

Citing 12
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Abstract

Ptolemaic graphs are graphs that satisfy the Ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs, and it is a natural generalization of block graphs (and hence trees). In this paper, a new characterization of ptolemaic graphs is presented. It is a laminar structure of cliques, and leads us to a canonical tree representation, which gives a simple intersection model for ptolemaic graphs. The tree representation is constructed in linear time from a perfect elimination ordering obtained by the lexicographic breadth first search. Hence the recognition and the graph isomorphism for ptolemaic graphs can be solved in linear time. Using the tree representation, we also give an O(n) time algorithm for the Hamiltonian cycle problem.