Maker-Breaker domination number

TL;DR

This paper introduces and analyzes the Maker-Breaker domination number in graph games, comparing it with classical domination parameters and providing bounds and exact values for specific graph classes.

Contribution

It defines the Maker-Breaker domination number, compares it with existing parameters, and determines it for trees, cycles, and graph unions, introducing residual graphs as a key tool.

Findings

 ext{MB} (G)  ext{ can be larger than the domination number  ext{ for certain graphs.

 ext{MB} (T) and  ext{MB} '(T) are explicitly determined for trees.

 ext{MB} (G) is bounded for unions of graphs.

Abstract

The Maker-Breaker domination game is played on a graph $G$ by Dominator and Staller. The players alternatively select a vertex of $G$ that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number $\gamma_{{\rm MB}}(G)$ of $G$ as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted $\gamma_{{\rm MB}}'(G)$. Comparing the two invariants it turns out that they behave much differently than the related game domination numbers. The invariant $\gamma_{{\rm MB}}(G)$ is also compared with the domination number. Using the Erd\H{o}s-Selfridge Criterion a large class of graphs $G$ is found for which $\gamma_{{\rm MB}}(G) > \gamma(G)$ holds. Residual graphs are introduced and used to bound/determine $\gamma_{{\rm MB}}(G)$ and $\gamma_{{\rm MB}}'(G)$. Using residual graphs, $\gamma_{{\rm MB}}(T)$ and $\gamma_{{\rm MB}}'(T)$ are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given.