将Robber限制在余图上
Confining the Robber on Cographs
摘要：
本文引入了图上的{ \em诱捕}和{ \em限制}的概念。对于某些$k\ge 4$，我们给出了图$G$不包含$k$顶点上的路（称为$P_k$ -free图）的结构必要条件，使得$k-3$警察没有策略来捕获或限制$G$上的强盗.利用这些条件，我们证明了平面上图和平面$P_5$ -free图的限制copnumber分别至多为1和2。它还表明，一个连接的上图，其中一个警察没有一个策略，以限制强盗的顶点数有一个紧下界为8。我们还探讨了双操作的影响-这是众所周知的，以提供一个特性的cograph-警察需要捕捉或限制抢劫犯的cograph上的数量。最后给出了关于无$P_5$图的限制copnumber和限制copnumber为3的最小可平面图的两个定理。
英文摘要：
In this paper, the notions of {\em trapping} and {\em confining} the robber on a graph are introduced. We present some structural necessary conditions for graphs $G$ not containing the path on $k$ vertices (referred to as $P_k$-free graphs) for some $k\ge 4$, so that $k-3$ cops do not have a strategy to capture or confine the robber on $G$. Utilizing such conditions, we show that for planar cographs and planar $P_5$-free graphs the confining cop number is at most one and two, respectively. It is also shown that the number of vertices of a connected cograph on which one cop does not have a strategy to confine the robber has a tight lower-bound of eight. We also explore the effects of twin operations -- which are well known to provide a characterization of cographs -- on the number of cops required to capture or confine the robber on cographs. We conclude by posing two conjectures concerning the confining cop number of $P_5$-free graphs and the smallest planar graph of confining cop number of three.
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。
