Maker-Breaker resolving game played on lexicographic products of graphs

TL;DR

This paper investigates the Maker-Breaker resolving game on lexicographic products of graphs, establishing winning strategies for players based on properties of the factor graphs and analyzing specific cases like paths and cycles.

Contribution

It extends the understanding of the Maker-Breaker resolving game to lexicographic product graphs, providing conditions for winning strategies and analyzing particular graph classes.

Findings

Spoiler wins on certain graph products regardless of starting player.

Resolver wins on products with even-length paths and cycles under specific conditions.

The Maker-Breaker resolving number is determined in most cases.

Abstract

In the Maker-Breaker resolving game, two players named Resolver and Spoiler alternately select unplayed vertices of a given graph $G$. The aim of Resolver is to select all the vertices of some resolving set of $G$, while Spoiler aims to select at least one vertex from every resolving set of $G$. In this paper, this game is investigated on the lexicographic product of graphs. It is proved that if Spoiler has a winning strategy on a graph $H$ no matter who starts the game, or if the first player has a winning strategy on $H$, then Spoiler always has a winning strategy on $G\circ H$. Special attention is paid to lexicographic products in which the second factor is either complete, or a path, or a cycle. For instance, in $G\circ P_{2\ell}$ and in $G\circ C_{2\ell}$, Resolver always wins, while in $G\circ P_{2\ell+1}$ and in $G\circ C_{2\ell+1}$ the same conclusion holds provided $G$ is free from false twins. On the other hand, Spoiler always wins on $G\circ P_5$. In most of the cases, the corresponding Maker-Breaker resolving number is also determined.