Maker-Breaker domination game

TL;DR

This paper introduces the Maker-Breaker domination game on graphs, analyzing its computational complexity and providing polynomial-time solutions for specific graph classes, along with strategies based on dominating set variants.

Contribution

It defines the Maker-Breaker domination game, proves PSPACE-completeness for general graphs, and offers polynomial algorithms and strategies for cographs and trees.

Findings

Deciding the winner is PSPACE-complete for bipartite and split graphs.

Polynomial algorithms exist for cographs and trees.

A strategy based on pairing dominating sets is proposed for Dominator.

Abstract

We introduce the Maker-Breaker domination game, a two player game on a graph. At his turn, the first player, Dominator, select a vertex in order to dominate the graph while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. This game is a particular instance of the so-called Maker-Breaker games, that is studied here in a combinatorial context. In this paper, we first prove that deciding the winner of the Maker-Breaker domination game is PSPACE-complete, even for bipartite graphs and split graphs. It is then showed that the problem is polynomial for cographs and trees. In particular, we define a strategy for Dominator that is derived from a variation of the dominating set problem, called the pairing dominating set problem.