arXiv 2007.159212020-07math.CO

关于笛卡儿积的局部化对策

The Localization Game On Cartesian Products

Jeandré Boshoff, Adriana Roux

备注：: 17 pages, 6 figures

摘要：

本地化游戏由两名玩家玩：一名警察和一队$k$警察，以及一名强盗。游戏初始化由强盗选择一个顶点$r \in V$，警察不知道。之后，游戏将以回合制进行。在每个回合开始时，Cop探测$k$顶点，并返回一个距离矢量。如果警察可以从矢量中确定$r$的确切位置，则强盗被定位，警察获胜。否则，强盗可以留在PDMATHAMO，或者移动到$r$附近的$r'$。然后Cop再次探测$k$顶点。游戏继续以这种方式进行，如果强盗可以在有限数量的回合中找到，警察就赢了。本地化数$ζ(G)$定义为最小正整数$k$，对于该最小正整数，无论强盗的移动如何，警察都具有获胜策略。在本文中，我们主要研究在笛卡儿积上的对策。我们证明了$ζ( G \square H) \geq \max\{ζ(G), ζ(H)\}$和$ζ(G \square H) \leq ζ(G) + ψ(H) - 1$，其中$ψ(H)$是$H$的一个双重可分解集.我们还表明，$ζ(C_m \square C_n)$是大多数等于2。

英文摘要：

The localization game is played by two players: a Cop with a team of $k$ cops, and a Robber. The game is initialised by the Robber choosing a vertex $r \in V$, unknown to the Cop. Thereafter, the game proceeds turn based. At the start of each turn, the Cop probes $k$ vertices and in return receives a distance vector. If the Cop can determine the exact location of $r$ from the vector, the Robber is located and the Cop wins. Otherwise, the Robber is allowed to either stay at $r$, or move to $r'$ in the neighbourhood of $r$. The Cop then again probes $k$ vertices. The game continues in this fashion, where the Cop wins if the Robber can be located in a finite number of turns. The localization number $ζ(G)$, is defined as the least positive integer $k$ for which the Cop has a winning strategy irrespective of the moves of the Robber. In this paper, we focus on the game played on Cartesian products. We prove that $ζ( G \square H) \geq \max\{ζ(G), ζ(H)\}$ as well as $ζ(G \square H) \leq ζ(G) + ψ(H) - 1$ where $ψ(H)$ is a doubly resolving set of $H$. We also show that $ζ(C_m \square C_n)$ is mostly equal to two.

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。

摘要：

英文摘要：

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