Patrolling cop vs omniscient robber

TL;DR

This paper investigates a pursuit-evasion game where a cop follows a fixed patrol and a fully informed robber aims to evade capture, analyzing the minimum capture radius needed across various graph classes.

Contribution

It introduces and systematically studies the parameter (G), providing exact values for trees, bounds for grids, and analysis for chordal graphs, advancing understanding of pursuit-evasion with fixed patrols.

Findings

Exact (G) for trees

Bounds for (G) on grids

Analysis of (G) for chordal graphs

Abstract

We study a variant of the classical Cops and Robbers game with one cop and one robber, in which the cop follows a fixed walk on the graph, a patrol, that is chosen before the game begins, while the robber is omniscient, he knows the entire patrol in advance. A capture occurs when the robber comes within a given radius of capture of the cop. This model arises naturally at the intersection of recent work on limited-visibility games and offline versions of pursuit-evasion problems. By $\tilde{\rho}{(G)}$ we denote the minimum radius of capture that the cop must have to always capture the robber on $G$ in this setting, under optimal play, where $G$ is a connected graph. We initiate a systematic study of this parameter for several graph classes. We determine the exact value of $\tilde{\rho}{(G)}$ 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.