Domination and matching in power and generalized power hypergraphs

Abstract

Let D be a subset of vertices of a hypergraph $${\mathcal {H}}$$. D is called 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 called the domination number of $${\mathcal {H}}$$, denoted by $$\gamma ({\mathcal {H}})$$. 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 $${\mathcal {H}}$$. It is known that $$\gamma ({\mathcal {H}})\le (r-1)\nu ({\mathcal {H}})$$ for any r-uniform hypergraph $${\mathcal {H}}$$. In this paper we investigate the relation between the domination number and matching number for some special hypergraphs. First, we prove that every power hypergraph H of rank r satisfies the inequality $$\nu (H)\le \gamma (H)\le 2\nu (H)$$, and we provide the complete characterizations of the power hypergraph H of rank r with $$\gamma (H)=\nu (H)$$ and $$\gamma (H)=2\nu (H)$$. Then we extend the corresponding results to generalized power hypergraphs. For any generalized power hypergraph $$H^{k,s}$$, we present $$\nu (H^{k,s})\le \gamma (H^{k,s})\le 2\nu (H^{k,s})$$ for $$1\le s< \frac{k}{2}$$ and $$\gamma (H^{k,s})\le \nu (H^{k,s})$$ for $$s= \frac{k}{2}$$. Furthermore, we completely characterize the extremal generalized power hypergraph $$H^{k,s}$$ with $$\gamma (H^{k,s})= \nu (H^{k,s})$$ and $$\gamma (H^{k,s})= 2\nu (H^{k,s})$$.