Geodesic
Clique graph representations of ptolemaic graphs
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 651-661

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: Ptolemaic graph, clique graph, chordal graph, clique tree, graph representation 
McKee, Terry. Clique graph representations of ptolemaic graphs. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 651-661. http://geodesic.mathdoc.fr/item/DMGT_2010_30_4_a9/
@article{DMGT_2010_30_4_a9,
     author = {McKee, Terry},
     title = {Clique graph representations of ptolemaic graphs},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {651--661},
     year = {2010},
     volume = {30},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2010_30_4_a9/}
}
TY  - JOUR
AU  - McKee, Terry
TI  - Clique graph representations of ptolemaic graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2010
SP  - 651
EP  - 661
VL  - 30
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/DMGT_2010_30_4_a9/
LA  - en
ID  - DMGT_2010_30_4_a9
ER  - 
%0 Journal Article
%A McKee, Terry
%T Clique graph representations of ptolemaic graphs
%J Discussiones Mathematicae. Graph Theory
%D 2010
%P 651-661
%V 30
%N 4
%U http://geodesic.mathdoc.fr/item/DMGT_2010_30_4_a9/
%G en
%F DMGT_2010_30_4_a9

[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).