Domination in intersecting hypergraphs

TL;DR

This paper investigates the structure of intersecting hypergraphs with maximum domination number, revealing that certain rank-4 hypergraphs with maximal domination are derived from the Fano plane.

Contribution

It characterizes the structure of intersecting hypergraphs with maximum domination number, especially those with rank 4, linking them to the Fano plane construction.

Findings

Intersects hypergraphs with rank r and maximum domination number have specific structural properties.

All linear intersecting hypergraphs of rank 4 with maximum domination are related to the Fano plane.

Provides a method to construct such hypergraphs based on known combinatorial designs.

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.