Partition strategies for the Maker-Breaker domination game

TL;DR

This paper investigates partition-based strategies for the Maker-Breaker domination game on various graph classes, providing polynomial-time algorithms and characterizations for Dominator's winning strategies.

Contribution

It introduces new partition strategies for Dominator, characterizes winning conditions in specific graph classes, and offers algorithms for complex graph subclasses.

Findings

Dominator always wins in regular graphs with certain partitions.

Deciding if Dominator wins as second player is polynomial-time for outerplanar and block graphs.

Existence of specific partitions characterizes winning strategies in interval graphs.

Abstract

The Maker-Breaker domination game is a positional game played on a graph by two players called Dominator and Staller. The players alternately select a vertex of the graph that has not yet been chosen. Dominator wins if at some point the vertices she has chosen form a dominating set of the graph. Staller wins if Dominator cannot form a dominating set. Deciding if Dominator has a winning strategy has been shown to be a PSPACE-complete problem even when restricted to chordal or bipartite graphs. In this paper, we consider strategies for Dominator based on partitions of the graph into basic subgraphs where Dominator wins as the second player. Using partitions into cycles and edges (also called perfect [1,2]-factors), we show that Dominator always wins in regular graphs and that deciding whether Dominator has a winning strategy as a second player can be computed in polynomial time for outerplanar and block graphs. We then study partitions into subgraphs with two universal vertices, which is equivalent to considering the existence of pairing dominating sets with adjacent pairs. We show that in interval graphs, Dominator wins if and only if such a partition exists. In particular, this implies that deciding whether Dominator has a winning strategy playing second is in NP for interval graphs. We finally provide an algorithm in $n^{k+3}$ for $k$-nested interval graphs (i.e. interval graphs with at most $k$ intervals included one in each other).