The minimum number of vertices in uniform hypergraphs with given domination number

TL;DR

This paper determines the minimum number of vertices in uniform hypergraphs with a specified domination number, providing asymptotic bounds and exploring related domination concepts.

Contribution

It establishes the asymptotic formula for the minimum vertices in hypergraphs with given domination number and extends the analysis to distance-$l$ domination.

Findings

Minimum vertices n(k,γ) = k + Θ(k^{1-1/γ})

Connected hypergraphs with distance-$l$ domination have roughly (kγl)/2 vertices

Results apply to s-wise and distance-$l$ domination variants

Abstract

The \textit{domination number} $\gamma(\mathcal{H})$ of a hypergraph $\mathcal{H}=(V(\mathcal{H}),E(\mathcal{H})$ is the minimum size of a subset $D\subset V(\mathcal{H}$ of the vertices such that for every $v\in V(\mathcal{H})\setminus D$ there exist a vertex $d \in D$ and an edge $H\in E(\mathcal{H})$ with $v,d\in H$. We address the problem of finding the minimum number $n(k,\gamma)$ of vertices that a $k$-uniform hypergraph $\mathcal{H}$ can have if $\gamma(\mathcal{H})\ge \gamma$ and $\mathcal{H}$ does not contain isolated vertices. We prove that $$n(k,\gamma)=k+\Theta(k^{1-1/\gamma})$$ and also consider the $s$-wise dominating and the distance-$l$ dominating version of the problem. In particular, we show that the minimum number $n_{dc}(k,\gamma, l)$ of vertices that a connected $k$-uniform hypergraph with distance-$l$ domination number $\gamma$ can have is roughly $\frac{k\gamma l}{2}$