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

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 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.

Keywords: Ptolemaic graph, clique graph, chordal graph, clique tree, graph representation.

2010 Mathematics Subject Classification: 05C62, 05C75.

References

[1]	H.-J. Bandelt and E. Prisner, Clique graphs and Helly graphs, J. Combin. Theory (B) 51 (1991) 34-45, doi: 10.1016/0095-8956(91)90004-4.
[2]	B.-L. Chen and K.-W. Lih, Diameters of iterated clique graphs of chordal graphs, J. Graph Theory 14 (1990) 391-396, doi: 10.1002/jgt.3190140311.
[3]	A. Brandstädt, V.B. Le and J.P. Spinrad, Graph Classes: A Survey, Society for Industrial and Applied Mathematics (Philadelphia, 1999).
[4]	E. Howorka, A characterization of ptolemaic graphs, J. Graph Theory 5 (1981) 323-331, doi: 10.1002/jgt.3190050314.
[5]	T.A. McKee, Maximal connected cographs in distance-hereditary graphs, Utilitas Math. 57 (2000) 73-80.
[6]	T.A. McKee and F.R. McMorris, Topics in Intersection Graph Theory, Society for Industrial and Applied Mathematics (Philadelphia, 1999).
[7]	F. Nicolai, A hypertree characterization of distance-hereditary graphs, Tech. Report Gerhard-Mercator-Universität Gesamthochschule (Duisburg SM-DU-255, 1994).
[8]	E. Prisner, Graph Dynamics, Pitman Research Notes in Mathematics Series #338 (Longman, Harlow, 1995).
[9]	J.L. Szwarcfiter, A survey on clique graphs, in: Recent advances in algorithms and combinatorics, pp. 109-136, CMS Books Math./Ouvrages Math. SMC 11 (Springer, New York, 2003).

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