On Helly Hypergraphs with Variable Intersection Sizes

A hypergraphH is said to be p-Helly when every p-wise intersecting partial hypergraph H′ of H has nonempty total intersection. Such hypergraphs have been characterized by Berge and Duchet in 1975, and since then they have appeared in the literature in several contexts, especially for the case p = 2, in which they are referred simply as Helly hypergraphs. An interesting generalization of p-Helly hypergraphs due to Voloshin takes into account not only the number of intersecting sets, but also the intersection sizes: we say that a hypergraph H is (p, q, s)-Helly when every p-wise q-intersecting partial hypergraph H′ of H has total intersection of cardinality at least s. In this work we propose a characterization for (p, q, s)-Helly hypergraphs. This characterization leads to an efficient algorithm to recognize such hypergraphs when p and q are fixed parameters.