On the generalized Helly property of hypergraphs, cliques, and bicliques

A family of sets is (p,q)-intersecting if every nonempty subfamily p or fewer has at least q elements in its total intersection. the (p,q)-Helly property intersection cardinality q. The (2,1)-Helly usual Helly property. hypergraph edge and hereditary each subhypergraphs graph (p,q)-clique-Helly maximal cliques induced subgraphs (p,q)-clique-Helly. classes (p,q)-biclique-Helly graphs are defined analogously. In this work, we prove several characterizations hypergraphs, including one by minimal forbidden partial subhypergraphs. On algorithmic side, give an improved time bound for recognition hypergraphs fixed show that can be solved polynomial co-NP-complete part input. addition, generalize characterization p-clique-Helly terms expansions to different graphs, subgraphs. We improvement on problem polynomial-time solvable NP-hard Finally, provide characterizations, algorithms, hardness results which analogous those graphs.