Bounds on the length of a game of Cops and Robbers

In the game of Cops and Robbers, a team of cops attempts to capture a robber on a graph G. All players occupy vertices of G. The game operates in rounds; in each round the cops move to neighboring vertices, after which the robber does the same. The minimum number of cops needed to guarantee capture of a robber on G is the cop number of G, denoted c(G), and the minimum number of rounds needed for them to do so is the capture time. It has long been known that the capture time of an n-vertex graph with cop number k is O(n). More recently, Bonato, Golovach, Hahn, and Kratochv́ıl ([3], 2009) and Gavenčiak ([9], 2010) showed that for k = 1, this upper bound is not asymptotically tight: for graphs with cop number 1, the cop can always win within n − 4 rounds. In this paper, we show that the upper bound is tight when k ≥ 2: for fixed k ≥ 2, we construct arbitrarily large graphs G having capture time at least (