Coarse cops and robber in graphs and groups
Louis Esperet, Harmender Gahlawat, and Ugo Giocanti

TL;DR
This paper studies two variants of the Cops and Robber game on infinite graphs and groups, establishing invariance under quasi-isometry, characterizations of cop numbers, and connections to graph minors and group properties.
Contribution
It introduces and analyzes weak and strong cop numbers in infinite graphs and groups, proving invariance under quasi-isometry and characterizing cases with cop number 1.
Findings
Weak and strong cop numbers are quasi-isometry invariants.
Graphs with cop number 1 are characterized.
The grid has infinite strong cop number.
Abstract
(abstract shortened to meet arxiv's length requirements) We investigate two variants of the classical Cops and robber game in graphs, recently introduced by Lee, Mart\'inez-Pedroza, and Rodr\'iguez-Quinche. The two versions are played in infinite graphs and the goal of the cops is to prevent the robber to visit some ball of finite radius (chosen by the robber) infinitely many times. Moreover the cops and the robber move at a different speed, and the cops can choose a radius of capture before the game starts. Depending on the order in which the parameters are chosen, this naturally defines two games, a weak version and a strong version (in which the cops are more powerful), and thus two variants of the cop number of a graph : the weak cop number and the strong cop number. It turns out that these two parameters are invariant under quasi-isometry and thus we can investigate these…
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