Domination and matching in power and generalized power hypergraphs

Yanxia Dong, Moo Young Sohn () and Zuosong Liang
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Yanxia Dong: Shanghai University of International Business and Economics
Moo Young Sohn: Mathematics, Changwon National University
Zuosong Liang: Qufu Normal University

Journal of Combinatorial Optimization, 2020, vol. 39, issue 2, No 8, 425-436

Abstract: Abstract Let D be a subset of vertices of a hypergraph $${\mathcal {H}}$$H. D is called a dominating set of $${\mathcal {H}}$$H if for every $$v\in V{\setminus } D$$v∈V\D there exists $$u\in D$$u∈D such that u and v lie in an hyperedge of $${\mathcal {H}}$$H. The cardinality of a minimum dominating set of $${\mathcal {H}}$$H is called the domination number of $${\mathcal {H}}$$H, denoted by $$\gamma ({\mathcal {H}})$$γ(H). A matching in a hypergraph $${\mathcal {H}}$$H is a set of pairwise disjoint hyperedges. The matching number $$\nu ({\mathcal {H}})$$ν(H) of $${\mathcal {H}}$$H is the size of a maximum matching in $${\mathcal {H}}$$H. It is known that $$\gamma ({\mathcal {H}})\le (r-1)\nu ({\mathcal {H}})$$γ(H)≤(r-1)ν(H) for any r-uniform hypergraph $${\mathcal {H}}$$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 $$\nu (H)\le \gamma (H)\le 2\nu (H)$$ν(H)≤γ(H)≤2ν(H), and we provide the complete characterizations of the power hypergraph H of rank r with $$\gamma (H)=\nu (H)$$γ(H)=ν(H) and $$\gamma (H)=2\nu (H)$$γ(H)=2ν(H). Then we extend the corresponding results to generalized power hypergraphs. For any generalized power hypergraph $$H^{k,s}$$Hk,s, we present $$\nu (H^{k,s})\le \gamma (H^{k,s})\le 2\nu (H^{k,s})$$ν(Hk,s)≤γ(Hk,s)≤2ν(Hk,s) for $$1\le s

Keywords: Hypergraph; Matching; Domination; Power hypergraph; Generalized power hypergraph; 05C65; 05C69; 05C70 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10878-019-00481-3

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