巡逻警察 vs 全知强盗
Patrolling cop vs omniscient robber
AI总结：
研究一名警察按预定巡逻路线行走、强盗全知并提前知晓路线的追逃博弈，定义巡逻捕获半径，并确定树、网格及弦图类的精确值或上下界。
摘要：
我们研究经典警察与强盗游戏的一个变种，其中有一名警察和一名强盗。警察在图上沿着一条固定的行走路径（称为巡逻路线）移动，该路径在游戏开始前选定。强盗是全知的，并且提前知道整个巡逻路线。当强盗进入警察周围给定距离内时发生捕获，该距离称为捕获距离。巡逻捕获半径 $\tilde{\rho}{(G)}$ 是在最优玩法下，警察在连通图 $G$ 上始终能够捕获强盗所需的最小捕获半径。我们针对几个图类系统研究了这个参数。我们确定了树的 $\tilde{\rho}{(G)}$ 精确值，建立了网格的上下界，并分析了包括区间图和毛虫图在内的各种弦图族的参数。在此过程中，我们开发了可能对研究具有预定巡逻和有限信息的追逃博弈具有独立兴趣的一般工具和结构结果。
英文摘要：
We study a variant of the classical Cops and Robbers game with one cop and one robber. The cop follows a fixed walk on the graph, called a patrol, that is chosen before the game begins. The robber is omniscient and knows the entire patrol in advance. A capture occurs when the robber comes within a given distance of the cop, and this distance is referred to as the capture distance. The patrol capture radius, $\tildeρ{(G)}$, is the minimum radius of capture required for the cop to always be able to capture the robber on a connected graph $G$, under optimal play. We initiate a systematic study of this parameter for several graph classes. We determine the exact value of $\tildeρ{(G)}$ for trees, establish upper and lower bounds for grids, and analyze the parameter for various families of chordal graphs, including interval graphs and caterpillars. Along the way, we develop general tools and structural results that may be of independent interest for the study of pursuit-evasion games with predetermined patrols and limited information.
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。
