Cop and Robber Games When the Robber Can Hide and Ride
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
... 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.