# Clique graph representations of ptolemaic graphs

Discussiones Mathematicae Graph Theory (2010)

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

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topTerry 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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- [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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