Maker-Breaker domination game critical graphs

TL;DR

This paper studies critical graphs in the Maker-Breaker domination game, characterizing those where the first player can win quickly, advancing understanding of strategic graph games.

Contribution

It establishes the existence of MBD game critical graphs and characterizes them for cases where the first player wins in one or two moves.

Findings

Critical graphs exist in the Maker-Breaker domination game.

Characterizations provided for cases with quick first-player wins.

Advances understanding of strategic graph game structures.

Abstract

The Maker-Breaker domination game (MBD game) is a two-player game played on a graph $G$ by Dominator and Staller. They alternately select unplayed vertices of $G$. The goal of Dominator is to form a dominating set with the set of vertices selected by him while that of Staller is to prevent this from happening. In this paper MBD game critical graphs are studied. Their existence is established and critical graphs are characterized for most of the cases in which the first player can win the game in one or two moves.