警察和强盗在网格和托里
Cops and robber on grids and tori
摘要：
本文是对图上经典的警察和强盗问题的一个贡献，针对二维网格和环形网格。这些研究通常旨在确定捕获抢劫犯所需的最少警察数量，并提出捕获算法。我们应用一些新的概念，提出了一个新的解决方案，网格上的问题，已经解决了一个不同的方法，并应用这些概念，给出有效的算法，环形网格上的捕获。至于网格，我们证明了两个警察足够，即使在半环面（即一个网格与环面封闭在一维）和三个警察是必要和充分的环面。然后，我们把问题的函数的任何数量k的警察，给出有效的算法，网格和环面和计算上下限的捕获时间。相反，我们确定任何给定的捕获时间所需的k的最小值，并研究可能的加速现象。
英文摘要：
This paper is a contribution to the classical cops and robber problem on a graph, directed to two-dimensional grids and toroidal grids. These studies are generally aimed at determining the minimum number of cops needed to capture the robber and proposing algorithms for the capture. We apply some new concepts to propose a new solution to the problem on grids that was already solved under a different approach, and apply these concepts to give efficient algorithms for the capture on toroidal grids. As for grids, we show that two cops suffice even in a semi-torus (i.e. a grid with toroidal closure in one dimension) and three cops are necessary and sufficient in a torus. Then we treat the problem in function of any number k of cops, giving efficient algorithms for grids and tori and computing lower and upper bounds on the capture time. Conversely we determine the minimum value of k needed for any given capture time and study a possible speed-up phenomenon.
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。
