The Disjoint Domination Game

TL;DR

This paper introduces a new Maker-Breaker game involving disjoint dominating sets in graphs, proving the maker's winning strategy in certain conditions and highlighting open problems for other game variants.

Contribution

It establishes the maker's winning strategy on all connected graphs when the breaker starts, and explores variants with biased and restricted rules.

Findings

Maker wins on all connected graphs if breaker starts

Maker can win on all graphs without isolated vertices in a restricted variant

Open problems remain for other game configurations

Abstract

We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game is started by the breaker. This implies the same in the $(2:1)$ biased game also in the maker-start game. It remains open to characterize the maker-win graphs in the maker-start non-biased game, and to analyze the $(a:b)$ biased game for $(a:b)\neq (2:1)$. For a more restricted variant of the non-biased game we prove that the maker can win on every graph without isolated vertices.