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arXiv · 2307.02373

Maker-Breaker Strong Resolving Game

Cong X. Kang·Aleksander Kelenc·Eunjeong Yi
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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.

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BibTeXRIS

Cong X. Kang, Aleksander Kelenc, Eunjeong Yi. 2023-07-05. Maker-Breaker Strong Resolving Game. https://arxiv.org/abs/2307.02373

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