arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2005.13242math.CO
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制造商-破坏者解决对策

Maker-Breaker resolving game

Cong X. Kang, Sandi Klavžar, Ismael G. Yero, Eunjeong Yi

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摘要：

一个图$G$的顶点集$W$是一个分解集，如果$G$的每个顶点都由它到$W$的距离向量唯一确定。本文介绍了庄家-庄家化解对策。游戏在图$G$上进行，由Resolver和Spoiler交替选择尚未选择的$G$的顶点。如果解析者选择的顶点在某个点形成$G$的解析集，则解析者获胜;如果解析者不能形成$G$的解析集，则剧透者获胜。游戏的结果分别用$o(G)$和$R_{\rm MB}(G)$表示。$S_{\rm MB}(G)$）表示解析器的最小移动次数（分别破坏者）赢得时，解析器有第一步。当剧透拥有第一步时，游戏的相应不变量由$R'_{\rm MB}(G)$和$S'_{\rm MB}(G)$表示。不变量$R_{\rm MB}(G)$、$R'_{\rm MB}(G)$、$S_{\rm MB}(G)$和$S'_{\rm MB}(G)$在它们自身之间以及与度量维度${\rm dim}(G)$进行比较。构造了一类图$G$，其中$R_{\rm MB}(G) > {\rm dim}(G)$成立.描述了孪生等价类和配对分解集对决策者-破坏者分解博弈的影响。作为一个应用程序$o(G)$，以及$R_{\rm MB}(G)$和$R'_{\rm MB}(G)$（或$S_{\rm MB}(G)$和$S'_{\rm MB}(G)$），确定了几个图形类，包括树，完全多部图，网格图，环面网格图。

英文摘要：

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 $R_{\rm MB}(G)$ (resp. $S_{\rm MB}(G)$) 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 $R'_{\rm MB}(G)$ and $S'_{\rm MB}(G)$. Invariants $R_{\rm MB}(G)$, $R'_{\rm MB}(G)$, $S_{\rm MB}(G)$, and $S'_{\rm MB}(G)$ are compared among themselves and with the metric dimension ${\rm dim}(G)$. A large class of graphs $G$ is constructed for which $R_{\rm MB}(G) > {\rm dim}(G)$ 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 $R_{\rm MB}(G)$ and $R'_{\rm MB}(G)$ (or $S_{\rm MB}(G)$ and $S'_{\rm MB}(G)$), are determined for several graph classes, including trees, complete multi-partite graphs, grid graphs, and torus grid graphs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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