Summary: 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 \(O(n^{k + 1})\). More recently, A. Bonato et al. [Discrete Math. 309, No. 18, 5588–5595 (2009; Zbl 1177.91056)] and T. Gavenčiak [Discrete Math. 310, No. 10–11, 1557–1563 (2010; Zbl 1186.91051)] 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 \(k \geq 2\): for fixed \(k \geq 2\), we construct arbitrarily large graphs \(G\) having capture time at least \(\left(\frac{\left|V(G)\right|}{40 k^4}\right)^{k + 1}\).
In the process of proving our main result, we establish results that may be of independent interest. In particular, we show that the problem of deciding whether \(k\) cops can capture a robber on a directed graph is polynomial-time equivalent to deciding whether \(k\) cops can capture a robber on an undirected graph. As a corollary of this fact, we obtain a relatively short proof of a major conjecture of A. S. Goldstein and E. M. Reingold [Theor. Comput. Sci. 143, No. 1, 93–112 (1995; Zbl 0873.68152)], which was recently proved through other means [W. B. Kinnersley, J. Comb. Theory, Ser. B 111, 201–220 (2015; Zbl 1307.05155)]. We also show that \(n\)-vertex strongly-connected directed graphs with cop number 1 can have capture time \(\varOmega(n^2)\), thereby showing that the result of A. Bonato et al. [loc. cit.] does not extend to the directed setting.