A Tight Lower Bound for the Capture Time of the Cops and Robbers Game
نویسندگان
چکیده
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(nk+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(nk+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
منابع مشابه
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 fo...
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