Progress towards the 1/2-Conjecture for the domination game

TL;DR

This paper advances the understanding of the domination game on graphs by providing a new upper bound on the number of moves needed for Dominator to finish, specifically for graphs with minimum degree at least 2.

Contribution

It proves a new upper bound on the game length for graphs with minimum degree at least 2, moving closer to the 1/2-Conjecture.

Findings

Dominator can finish the game within approximately 10n/17 moves.

Progress made towards the 1/2-Conjecture for graphs with minimum degree ≥ 2.

Provides a strategic approach for Dominator to minimize game length.

Abstract

The domination game is played on a graph $G$ by two players, Dominator and Staller, who alternate in selecting vertices until each vertex in the graph $G$ is contained in the closed neighbourhood of the set of selected vertices. Dominator's aim is to reach this state in as few moves as possible, whereas Staller wants the game to last as long as possible. In this paper, we prove that if $G$ has $n$ vertices and minimum degree at least 2, then Dominator has a strategy to finish the domination game on $G$ within $10n/17+1/17$ moves, thus making progress towards a conjecture by Bujt{\'a}s, Ir\v{s}i\v{c} and Klav\v{z}ar.