Characterization and recognition of generalized clique-Helly graphs

Authors:
Mitre C. Dourado;Fábio Protti;Jayme L. Szwarcfiter
Affiliations:
COPPE, Sistemas, Brazil;IM and NCE, Caixa Postal 2324, 20001-970 Universidade Federal do Rio de Janeiro Rio de Janeiro, RJ, Brazil;COPPE, Sistemas, Brazil and IM and NCE, Caixa Postal 2324, 20001-970 Universidade Federal do Rio de Janeiro Rio de Janeiro, RJ, Brazil
Venue:
Discrete Applied Mathematics
Year:
2007

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Abstract

Let p=1 and q=0 be integers. A family of sets F is (p,q)-intersecting when every subfamily F^'@?F formed by p or less members has total intersection of cardinality at least q. A family of sets F is (p,q)-Helly when every (p,q)-intersecting subfamily F^'@?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.