图的损伤数
On the damage number of graphs
浏览论文内容
中文总结 AI 辅助
本文研究警察与强盗博弈的变体，确定超立方体等多类图的损伤数，并证明确定图损伤数的问题是PSPACE完全的。
中文摘要 AI 辅助
我们研究了警察与强盗博弈的一个变体：强盗试图在不被捕获的情况下尽可能多地访问图的顶点，而警察的目标是将强盗限制在一小部分顶点内。图G的损伤数（damage number）由Cox和Sanaei于2019年提出，指的是在图G上进行的警察与强盗博弈中，强盗能访问的最大顶点数。本文确定了几类图的损伤数，包括超立方体、汉明图、约翰逊图、射影平面的关联图，以及满足p≫log^(2/5)(n)/n^(1/5)的Erdős–Rényi随机图G(n,p)。我们还证明了确定图的损伤数的问题是PSPACE完全的。
英文摘要
We study a variant of Cops and Robbers in which the robber attempts to visit as many vertices of the graph as possible without being captured, while the cop aims to keep the robber confined to a small set of vertices. The \textit{damage number} of a graph $G$, introduced by Cox and Sanaei in 2019, is the maximum number of vertices the robber can visit in a game of Cops and Robbers on $G$. In this paper, we determine damage numbers for several classes of graphs, including hypercubes, Hamming graphs, Johnson graphs, incidence graphs of projective planes, and Erdős-Rényi random graph $G(n,p)$, for $p \gg \log^{2/5}(n) / n^{1/5}$. We also show that the problem of determining the damage number of a graph is {\sf PSPACE}-complete.
