A matching in a hypergraph $\mathcal{ H}$ is a set of pairwise disjoint hyperedges. The matching number $\nu(\mathcal{ H})$ of $\mathcal{ H}$ is the size of a maximum matching in $\mathcal{ H}$. A subset $D$ of vertices of $\mathcal{ H}$ is a dominating set of $\mathcal{ H}$ if for every $v\in V\setminus D$ there exists $u\in D$ such that $u$ and $v$ lie in an hyperedge of $\mathcal{ H}$. The cardinality of a minimum dominating set of $\mathcal{ H}$ is the domination number of $\mathcal{ H}$, denoted by $\gamma(\mathcal{ H})$. It was proved that $\gamma(\mathcal{ H})\leq (r-1)\nu(\mathcal{ H})$ for $r$-uniform hypergraphs and the 2-uniform hypergraphs (graphs) achieving equality $\gamma(\mathcal{ H})=\nu(\mathcal{ H})$ have been characterized. In this paper we generalize the inequality $\gamma(\mathcal{ H})\leq (r-1)\nu(\mathcal{ H})$ to arbitrary hypergraph of rank $r$ and we completely characterize the extremal hypergraphs $\mathcal{ H}$ of rank $3$ achieving equality $\gamma(\mathcal{ H})=(r-1)\nu(\mathcal{ H})$.
Categories: math.CO