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arXiv · 1701.01564

Domination in intersecting hypergraphs

Yanxia Dong·Erfang Shan·Shan Li·Liying Kang
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Abstract

A matching in a hypergraph $H$ is a set of pairwise disjoint hyperedges. The matching number $\alpha'(H)$ of $H$ is the size of a maximum matching in $H$. A subset $D$ of vertices of $H$ is a dominating set of $H$ if for every $v\in V\setminus D$ there exists $u\in D$ such that $u$ and $v$ lie in an hyperedge of $H$. The cardinality of a minimum dominating set of $H$ is called the domination number of $H$, denoted by $\gamma(H)$. It is known that for a intersecting hypergraph $H$ with rank $r$, $\gamma(H)\leq r-1$. In this paper we present structural properties on intersecting hypergraphs with rank $r$ satisfying the equality $\gamma(H)=r-1$. By applying the properties we show that all linear intersecting hypergraphs $H$ with rank $4$ satisfying $\gamma(H)=r-1$ can be constructed by the well-known Fano plane.

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Yanxia Dong, Erfang Shan, Shan Li, Liying Kang. 2017-01-06. Domination in intersecting hypergraphs. https://arxiv.org/abs/1701.01564

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