Equality of domination and transversal numbers in hypergraphs

Abstract

A subset S of the vertex set of a hypergraph H is called a dominating set of H if for every vertex v not in S there exists u∈S such that u and v are contained in an edge in H. The minimum cardinality of a dominating set in H is called the domination number of H and is denoted by γ(H). A transversal of a hypergraph H is defined to be a subset T of the vertex set such that T∩E≠0̸ for every edge E of H. The transversal number of H, denoted by τ(H), is the minimum number of vertices in a transversal. A hypergraph is of rank k if each of its edges contains at most k vertices.The inequality τ(H)≥γ(H) is valid for every hypergraph H without isolated vertices. In this paper, we investigate the hypergraphs satisfying τ(H)=γ(H), and prove that their recognition problem is NP-hard already on the class of linear hypergraphs of rank 3, while on unrestricted problem instances it lies inside the complexity class Θ2p.Structurally we focus our attention on hypergraphs in which each subhypergraph H′ without isolated vertices fulfills the equality τ(H′)=γ(H′). We show that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, we prove that 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.