A set of vertices W of a graph G is a resolving set if every vertex of G is uniquely determined by its vector of distances to W. In this paper, the Maker–Breaker resolving game is introduced. The game is played on a graph G by Resolver and Spoiler who alternately select a vertex of G not yet chosen. Resolver wins if at some point the vertices chosen by him form a resolving set of G, whereas Spoiler wins if the Resolver cannot form a resolving set of G. The outcome of the game is denoted by o(G), and \(R_{\mathrm{MB}}(G)\) (resp. \(S_{\mathrm{MB}}(G)\)) denotes the minimum number of moves of Resolver (resp. Spoiler) to win when Resolver has the first move. The corresponding invariants for the game when Spoiler has the first move are denoted by \(R'_{\mathrm{MB}}(G)\) and \(S'_\mathrm{MB}(G)\). Invariants \(R_{\mathrm{MB}}(G)\), \(R'_{\mathrm{MB}}(G)\), \(S_\mathrm{MB}(G)\), and \(S'_{\mathrm{MB}}(G)\) are compared among themselves and with the metric dimension \(\mathrm{dim}(G)\). A large class of graphs G is constructed for which \(R_{\mathrm{MB}}(G) > \mathrm{dim}(G)\) holds. The effect of twin equivalence classes and pairing resolving sets on the Maker–Breaker resolving game is described. As an application, o(G), as well as \(R_{\mathrm{MB}}(G)\) and \(R'_{\mathrm{MB}}(G)\) (or \(S_{\mathrm{MB}}(G)\) and \(S'_{\mathrm{MB}}(G)\)), is determined for several graph classes, including trees, complete multi-partite graphs, grid graphs, and torus grid graphs.

如果G的每个顶点由其到W的距离向量唯一地确定，则图G的一组顶点W是一个解集。本文介绍了Maker-Breaker解决游戏。游戏由Resolver和Spoiler在图G上进行，他们交替选择一个尚未选择的G顶点。解析器胜如果在某一时刻由他所选择的顶点形成的分辨组G ^，而扰流板赢，如果解析器不能形成拆分组G ^。游戏的结果用o（G）和\（R _ {\ mathrm {MB}}（G）\）（resp。\（S _ {\ mathrm {MB}}（G）\））表示当旋转变压器第一步移动时获胜的最小旋转变压器（resp。Spoiler）数量。当Spoiler出手时，游戏的相应不变量由\（R'_ {\ mathrm {MB}}（G）\）和\（S'_ \ mathrm {MB}（G）\）表示。不变量\（R _ {\ mathrm {MB}}（G）\），\（R'_ {\ mathrm {MB}}（G）\），\（S_ \ mathrm {MB}（G）\）和\（S'_ {\ mathrm {MB}}（G）\）之间相互比较，并与度量维度\（\ mathrm {dim}（G）\）比较。构造了一大类图G，其中\（R _ {\ mathrm {MB}}（G）> \ mathrm {dim}（G）\）持有。描述了双对等类和配对解析集对Maker-Breaker解析游戏的影响。作为应用程序，o（G）以及\（R _ {\ mathrm {MB}}（G）\）和\（R'_ {\ mathrm {MB}}（G）\）（或\（S_ {\ mathrm {MB}}（G）\）和\（S'_ {\ mathrm {MB}}（G）\））被确定用于几个图类，包括树，完整的多部分图，网格图和环面网格图。