The Robber Locating game

In 2012 Seager introduced a new variant of the Cops and Robbers game, in which a cop searches for a moving robber on a graph using distance probes. The robber on his turn can either remain still or move to an adjacent vertex, while the cop on her turn probes any vertex and learns the robber’s distance from the probed vertex. Carraher, Choi, Delcourt, Erickson and West later showed that for any fixed graph G there is a winning strategy for the cop on the graph G, obtained by replacing each edge of G by a path of length m, if m is sufficiently large. They conjectured that the cop does not have a winning strategy on K 1/m n if m < n; we show that in fact the cop wins if and only if m > n/2, for all but a few small values of n. We also give a complete answer to the question of when the cop has a winning strategy on K 1/m a,b .