Clique graph representations of ptolemaic graphs

• Autorzy: Terry A. Mckee
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Ptolemaic graph; clique graph; chordal graph; clique tree; graph representation

Kategorie tematyczne

• 05C62: Graph representations (geometric and intersection representations, etc.) {For graph drawing, see also }
• 05C75: Structural characterization of families of graphs

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2010
• Tom: 30
• Numer: 4
• Strony: 651-661

Daty

wydano

2010

otrzymano

2009-04-17

poprawiono

2010-02-23

zaakceptowano

2010-03-02

Twórcy

autor

Terry A. Mckee

• Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435, USA

Bibliografia

• [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).

Identyfikatory

DOI

10.7151/dmgt.1520