# Clique graph representations of ptolemaic graphs

• Volume: 30, Issue: 4, page 651-661
• ISSN: 2083-5892

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## Abstract

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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 ${P}_{n+1}$-free induced subgraphs of G in which every two nonadjacent vertices are connected by at least p internally disjoint paths.

## How to cite

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Terry A. Mckee. "Clique graph representations of ptolemaic graphs." Discussiones Mathematicae Graph Theory 30.4 (2010): 651-661. <http://eudml.org/doc/271064>.

@article{TerryA2010,
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 $P_\{n+1\}$-free induced subgraphs of G in which every two nonadjacent vertices are connected by at least p internally disjoint paths.},
author = {Terry A. Mckee},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {Ptolemaic graph; clique graph; chordal graph; clique tree; graph representation},
language = {eng},
number = {4},
pages = {651-661},
title = {Clique graph representations of ptolemaic graphs},
url = {http://eudml.org/doc/271064},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Terry A. Mckee
TI - Clique graph representations of ptolemaic graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 4
SP - 651
EP - 661
AB - 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 $P_{n+1}$-free induced subgraphs of G in which every two nonadjacent vertices are connected by at least p internally disjoint paths.
LA - eng
KW - Ptolemaic graph; clique graph; chordal graph; clique tree; graph representation
UR - http://eudml.org/doc/271064
ER -

## References

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1. [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. Zbl0726.05060
2. [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. Zbl0726.05059
3. [3] A. Brandstädt, V.B. Le and J.P. Spinrad, Graph Classes: A Survey, Society for Industrial and Applied Mathematics (Philadelphia, 1999).
4. [4] E. Howorka, A characterization of ptolemaic graphs, J. Graph Theory 5 (1981) 323-331, doi: 10.1002/jgt.3190050314. Zbl0437.05046
5. [5] T.A. McKee, Maximal connected cographs in distance-hereditary graphs, Utilitas Math. 57 (2000) 73-80. Zbl0953.05024
6. [6] T.A. McKee and F.R. McMorris, Topics in Intersection Graph Theory, Society for Industrial and Applied Mathematics (Philadelphia, 1999). Zbl0945.05003
7. [7] F. Nicolai, A hypertree characterization of distance-hereditary graphs, Tech. Report Gerhard-Mercator-Universität Gesamthochschule (Duisburg SM-DU-255, 1994).
8. [8] E. Prisner, Graph Dynamics, Pitman Research Notes in Mathematics Series #338 (Longman, Harlow, 1995).
9. [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). Zbl1027.05071

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