We study a game on a graph played by revolutionaries and spies. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move, and then each spy has the same option. The revolutionaries win if of them meet at some vertex having no spy (at the end of a round); the spies win if they can avoid this forever.
Let denote the minimum number of spies needed to win. To avoid degenerate cases, assume . The easy bounds are then . We prove that the lower bound is sharp when has a rooted spanning tree such that every edge of not in joins two vertices having the same parent in . As a consequence, , where is the domination number; this bound is nearly sharp when .
For the random graph with constant edge-probability , we obtain constants and (depending on and ) such that is near the trivial upper bound when and at most times the trivial lower bound when . For the hypercube with , we have when , and for at least spies are needed.
For complete -partite graphs with partite sets of size at least , the leading term in is approximately when . For , we have and , and in general .