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Bounds on the length of a game of cops and robbers. (English) Zbl 1392.05082
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.

##### MSC:
 05C57 Games on graphs (graph-theoretic aspects) 91A43 Games involving graphs 91A24 Positional games (pursuit and evasion, etc.)
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