Maker-Breaker Strong Resolving Game

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.