Characterization and Recognition of Generalized Clique-Helly Graphs

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.