Domination game on uniform hypergraphs

TL;DR

This paper introduces and analyzes the domination game on hypergraphs, establishing bounds on the game domination number for uniform hypergraphs and related graph classes, extending previous graph results to hypergraph settings.

Contribution

It generalizes the domination game to hypergraphs, provides asymptotically tight bounds for k-uniform hypergraphs, and improves bounds for specific graph classes.

Findings

Established an asymptotically tight upper bound for k-uniform hypergraphs.

Proved that for 3-uniform hypergraphs without isolated vertices, the game domination number is at most 5n/9.

Extended the bound to graphs where each edge is in a triangle, showing the same limit applies.

Abstract

In this paper we introduce and study the domination game on hypergraphs. This is played on a hypergraph $\mathcal{H}$ by two players, namely Dominator and Staller, who alternately select vertices such that each selected vertex enlarges the set of vertices dominated so far. The game is over if all vertices of $\mathcal{H}$ are dominated. Dominator aims to finish the game as soon as possible, while Staller aims to delay the end of the game. If each player plays optimally and Dominator starts, the length of the game is the invariant `game domination number' denoted by $\gamma_g(\mathcal{H})$. This definition is the generalization of the domination game played on graphs and it is a special case of the transversal game on hypergraphs. After some basic general results, we establish an asymptotically tight upper bound on the game domination number of $k$-uniform hypergraphs. In the remaining part of the paper we prove that $\gamma_g(\mathcal{H}) \le 5n/9$ if $\mathcal{H}$ is a 3-uniform hypergraph of order $n$ and does not contain isolated vertices. This also implies the following new result for graphs: If $G$ is an isolate-free graph on $n$ vertices and each of its edges is contained in a triangle, then $\gamma_g(G) \le 5n/9$.