A cop-winning strategy on strongly cop-win graphs.

Nowakowski and Winkler [5] introduced a game on graphs where a cop tries to catch a robber; in particular they showed that the cop has a winning strategy on a (finite or infinite) graph if and only if the graph is dismantlable. In this paper, we prove a similar result for a game on a finite graph where the robber can travel through arbitrary many edges in a single move. The original dismantling procedure consists in deleting specific vertices one by one until reaching a single vertex. This procedure fails with a fast robber because by deleting vertices we remove possible escape ways for the robber, and hence possibly change the nature of the graph from robber-win to cop-win. To overcome this problem, we define a new dedicated light-dismantling procedure, which is a generalization of the classical one, with a broader application possibility, not only to the fast robber game. Instead of being deleted, vertices are darkened: dark vertices can be only visited by the robber, in passing. Hence we introduce undergrounded graphs, on which the game will be played. We define a monotonic winning strategy for the cop that ensures that she is progressing towards the robber, thus securing a growing area.