Cliques and extended triangles. A necessary condition for planar clique graphs

Authors:
Liliana Alcón;Marisa Gutierrez
Affiliations:
Departamento Matemática, Universidad Nacional de La Plata, C.C. 172, 1900 La Plata, Argentina;Departamento Matemática, Universidad Nacional de La Plata, C.C. 172, 1900 La Plata, Argentina
Venue:
Discrete Applied Mathematics - Brazilian symposium on graphs, algorithms and combinatorics
Year:
2004

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Abstract

By generalizing the idea of extended triangle of a graph, we succeed in obtaining a common framework for the result of Roberts and Spencer about clique graphs and the one of Szwarcfiter about Helly graphs. We characterize Helly and 3-Helly planar graphs using extended triangles. We prove that if a planar graph G is a clique graph, then every extended triangle of G must be a clique graph. Finally, we show the extended triangles of a planar graph which are clique graphs. Any one of the obtained characterizations are tested in O(n2) time.