Maker-Breaker Strong Resolving Game

TL;DR

This paper introduces a new game-theoretic approach to strong resolving sets in graphs, analyzing outcomes and strategies for Maker and Breaker across various graph classes.

Contribution

It defines the maker-breaker strong resolving game, explores its properties, and determines outcomes for specific graph classes, advancing understanding of graph resolving strategies.

Findings

Established general results on MBSRG outcomes.

Analyzed the relation between MBSRG and the resolving game.

Determined outcomes for corona, Cartesian, and modular product graphs.

Abstract

Let $G$ be a graph with vertex set $V$. A set $S \subseteq V$ is a \emph{strong resolving set} of $G$ if, for distinct $x,y\in V$, there exists $z\in S$ such that either $x$ lies on a $y-z$ geodesic or $y$ lies on an $x-z$ geodesic in $G$. In this paper, we study maker-breaker strong resolving game (MBSRG) played on a graph by two players, Maker and Breaker, where the two players alternately select a vertex of $G$ not yet chosen. Maker wins if he is able to choose vertices that form a strong resolving set of $G$ and Breaker wins if she is able to prevent Maker from winning in the course of MBSRG. We denote by $O_{\rm SR}(G)$ the outcome of MBSRG played on $G$. We obtain some general results on MBSRG and examine the relation between $O_{\rm SR}(G)$ and $O_{\rm R}(G)$, where $O_{\rm R}(G)$ denotes the outcome of the maker-breaker resolving game of $G$. We determine the outcome of MBSRG played on some graph classes, including corona product graphs, Cartesian product graphs, and modular product graphs.