  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. 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 graphs admitting a dismantling scheme. In this paper, we
characterize in a similar way the class CW(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 CW(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 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 cop-win for any value of k. Finally, we
consider the game where the cop wins if he is at distance at most 1 from the
robber and we characterize via a specific dismantling scheme the bipartite
graphs where a single cop wins in this game.
