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