Cop and Robber Games When the Robber Can Hide and Ride

Jérémie Chalopin, Victor Chepoi, Nicolas Nisse, Yann Vaxès
2011 SIAM Journal on Discrete Mathematics  
In the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G = (V, E). The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber after a finite number of steps. Nowakowski, Winkler (1983) and Quilliot (1983) characterized the cop-win graphs as dismantlable graphs. In this talk, we will characterize in a similar way the class
more » ... WFR(s, s ) of cop-win graphs in the game in which the cop and the robber move at different speeds s and s, s ≤ s. We also establish some connections between cop-win graphs for this game with s < s and Gromov's hyperbolicity. In the particular case s = 1 and s = 2, we prove that the class of cop-win graphs is exactly the well-known class of dually chordal graphs. We show that all classes CWFR(s, 1), s ≥ 3, coincide and we provide a structural characterization of these graphs. We also investigate several dismantling schemes necessary or sufficient for the cop-win graphs (which we call k-winnable and denote by CWW(k)) in the game in which the robber is visible only every k moves for a fixed integer k > 1. We characterize the graphs which are k-winnable for any value of k. We present now our main results (for the full version, see . A graph G is dually chordal if its clique hypergraph is a hypertree. Dually chordal graphs are exactly the graphs G = (V, E) admitting a maximum neighborhood ordering.
doi:10.1137/100784035 fatcat:j7hm3mnjwnerpfr6jeydniiuzm