当前位置: X-MOL 学术arXiv.cs.DM › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Confining the Robber on Cographs
arXiv - CS - Discrete Mathematics Pub Date : 2020-06-16 , DOI: arxiv-2006.08941
Masood Masjoody

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.

中文翻译:

将强盗限制在 Cographs 上

在本文中,介绍了{\em 捕获} 和{\em 限制} 图上的强盗的概念。我们为某些 $k\ge 4$ 的图 $G$ 不包含 $k$ 顶点上的路径(称为 $P_k$-free 图)提供一些结构必要条件,以便 $k-3$ cops 不包含有一个策略来捕获或限制 $G$ 的强盗。利用这些条件,我们表明对于平面 cographs 和平面 $P_5$-free 图,限制 cop 数最多分别为 1 和 2。还表明,一个警察没有限制强盗的策略的连通图的顶点数量有一个紧的下限 8。我们还探讨了双操作的影响——众所周知,它可以提供 cograph 的特征——对捕获或限制 cograph 上的强盗所需的警察数量的影响。
更新日期:2020-09-15
down
wechat
bug