A tight lower bound for the capture time of the Cops and Robbers game

Abstract

For the game of Cops and Robbers, it is known that in 1-cop-win graphs, the cop can capture the robber in O(n) time, and that there exist graphs in which this capture time is tight. When k ≥ 2, a simple counting argument shows that in k-cop-win graphs, the capture time is at most O(n k+1), however, no non-trivial lower bounds were previously known; indeed, in their 2011 book, Bonato and Nowakowski ask whether this upper bound can be improved. In this paper, the question of Bonato and Nowakowski is answered on the negative, proving that the O(n k+1) bound is asymptotically tight for any constant k ≥ 2. This yields a surprising gap in the capture time complexities between the 1-cop and the 2-cop cases.

Topological directions in Cops and Robbers

“…Recent work by Kinnersley [25] and independently by Brandt et al [15] shows that there exist families of graphs with cop number k such that capt k (G) = Θ(n c(G)+1 ).…”

Lower Bound of Zero-visibility Cops and Robber Game on the Cube Grid

“…In these three models, the robber knows the location of all cops at any time, but the cop's visibility of the robber is different. The first is that the cops know the perfect information of the robber, a lot of results can be found in [3,4,5,6]. The second is that the cops know the imperfect information of the robber.…”

The Game of Cops and Robber on (Claw, Even-hole)-free Graphs

“…However, surprisingly, the upper bound happens to be asymptotically tight for all integers k ≥ 2, i.e. there are k-cop-win graphs with the capture time in Θ(n k+1 ) for every integer k ≥ 2 [Kin18,BEUW20].…”