Cops vs. Gambler

We consider a variation of cop vs.\ robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on $V(G)$ and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We show that when the gambler's distribution is known, the expected capture time (with best play) on any connected $n$-vertex graph is exactly $n$. We also give bounds on the (generally greater) expected capture time when the gambler's distribution is unknown to the cop.Comment: 6 pages, 0 figure

The capture time of grids

We consider the game of Cops and Robber played on the Cartesian product of two trees. Assuming the players play perfectly, it is shown that if there are two cops in the game, then the length of the game (known as the 2-capture time of the graph) is equal to half the diameter of the graph. In particular, the 2-capture time of the m x n grid is proved to be floor ((m+n-2)/2).Comment: 7 page