Equality of domination and transversal numbers in hypergraphs

The domination number γ(H) and the transversal number τ(H) (also called vertex covering number) of a hypergraph H are defined analogously to those of a graph. A hypergraph is of rank k if each edge contains at most k vertices. The inequality τ(H) ≥ γ(H) is valid for every hypergraph H without isolated vertices. We study the structure of hypergraphs satisfying τ(H) = γ(H), moreover prove that the corresponding recognition problem is NP-hard already on the class of linear hypergraphs of rank 3. We focus our attention mostly on hypergraphs for which τ = γ hereditarily holds, that is in which each subhypergraph H′ without isolated vertices fulfills the equality τ(H′) = γ(H′). We prove that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, for every positive integer k, there are only a finite number of forbidden subhypergraphs of rank k, and each of them has domination number at most k. Thus, hypergraphs for which τ = γ hereditarily holds can be recognized in polynomial time if the rank is fixed.