Characterization and recognition of generalized clique-helly graphs

  • Authors:

  • Mitre Costa Dourado;Fábio Protti;Jayme Luiz Szwarcfiter

  • Affiliations:

  • COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brasil;Instituto de Matemática and Núcleo de Computação Eletrônica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brasil;Instituto de Matemática, Núcleo de Computação Eletrônica and COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brasil

  • Venue:
  • WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
  • Year:
  • 2004

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Abstract

Let p ≥ 1 and q ≥ 0 be integers. A family of sets ${\mathcal F}$ is (p,q)-intersecting when every subfamily ${\mathcal F}' \subseteq {\mathcal F}$ formed by p or less members has total intersection of cardinality at least q. A family of sets ${\mathcal F}$ is (p,q)-Helly when every (p,q)-intersecting subfamily ${\mathcal F}' \subseteq {\mathcal F}$ has total intersection of cardinality at least q. A graph G is a (p,q)-clique-Helly graph when its family of (maximal) cliques is (p,q)-Helly. According to this terminology, the usual Helly property and the clique-Helly graphs correspond to the case p=2, q=1. In this work we present a characterization for (p,q)-clique-Helly graphs. For fixed p,q, this characterization leads to a polynomial-time recognition algorithm. When p or q is not fixed, it is shown that the recognition of (p,q)-clique-Helly graphs is NP-hard.