Coarse cops and robber games

Ugo Giocanti
Jagiellonian University, Kraków
https://ugoue.github.io/

Date(s) : 06/01/2026 iCal
11h00 - 12h00

Joint work with Louis Esperet and Harmender Gahlawat.

Lee, Martínez-Pedroza, and Rodríguez-Quinche recently introduced two coarse variants of the classical Cops and robber game in inﬁnite graphs. In these two variants, ﬁnitely many cops want to prevent a robber to enter inside a ball of ﬁnite radius infinitely often. Both the cops and the robber have some speeds, and the cops also have some radius of capture. According to the order in which all these parameters are chosen, we can deﬁne two diﬀerent games, a weak and a strong version. These games allow deﬁne two parameters for inﬁnite graphs, respectively called weak and strong cop numbers.

An important property of these parameters proved by Lee, Martínez-Pedroza and Rodríguez-Quinche, is that they are invariant under quasi-isometries. I will present a number of results and questions about these two parameters. In particular, I will show how the weak cop number of a graph is related to the existence of some asymptotic minors of large treewidth, establishing some connections with some recent question by Georgakopoulos and Papasoglou in coarse graph theory. I will also show that graphs with strong cop number 1 are exactly the hyperbolic graphs (the converse direction was proved by Lee, Martínez-Pedroza and Rodríguez-Quinche), which complements in some way some previous results of Chalopin, Chepoi, Nisse, Papasoglou, Pecatte and Vaxès in ﬁnite graphs. Eventually, I will show some applications of these results when one works on ﬁnitely generated groups.