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

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... 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. 1998 ACM Subject Classification G.2.2 Graph Theory 1 Introduction The game of Cops and Robbers is a perfect information two-player zero-sum game played on an undirected n-vertex graph G = (V, E), where the first player is identified with k ≥ 1 cops, indexed by the integers 0,. .. , k − 1, and the second player is identified with a single robber. In round 0, the cop player chooses the initial (not necessarily distinct) cop locations c 0 (0),. .. , c k−1 (0) ∈ V and following that, the robber player chooses the initial robber location r(0) ∈ V. Then, in round t = 1, 2,. .. , the cop player chooses the next (not necessarily distinct) cop locations c 0 (t),. .. , c k−1 (t) ∈ V under the constraint that c i (t) ∈ N + (c i (t − 1)) for every 0 ≤ i ≤ k − 1, where N + (v) denotes the neighborhood of vertex v in G including v itself; following that, the robber player chooses the next robber location r(t) ∈ N + (r(t − 1)). The goal of the cop player is to ensure that r(t − 1) ∈ {c 0 (t),. .. , c k−1 (t)} for some finite round t, referred to as capturing the robber. Conversely, the goal of the robber player is to * A full version of this paper can be found at