We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager (Seager, 2012). Carragher, Choi, Delcourt, Erickson and West showed that for any n-vertex graph G there is a winning strategy for the cop on the graph G 1/m obtained by replacing each edge of G by a path of length m, if m ≥ n (Carragher et al., 2012). The present authors showed that, for all but a few small values of n, this bound may be improved to m ≥ n/2, which is best possible (Haslegrave et al., 2016b). In this paper we consider the natural extension in which the cop probes a set of k vertices, rather than a single vertex, at each turn. We consider the relationship between the value of k required to ensure victory on the original graph with the length of subdivisions required to ensure victory with k = 1. We give an asymptotically best-possible linear bound in one direction, but show that in the other direction no subexponential bound holds. We also give a bound on the value of k for which the cop has a winning strategy on any (possibly infinite) connected graph of maximum degree ∆, which is best possible up to a factor of (1 − o(1)).

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We consider a game in which a cop searches for a moving robber on a graph using distance probes, studied by Carragher, Choi, Delcourt, Erickson and West, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West show that for any fixed graph G there is a winning strategy for the cop on the graph G 1/m , obtained by replacing each edge of G by a path of length m, if m is sufficiently large. They conjecture 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. They also show that the robber can avoid capture on any graph of girth 3, 4 or 5, and ask whether there is any graph of girth 6 on which the cop wins. We show that there is, but that no such graph can be bipartite; in the process we give a counterexample for their conjecture that the set of graphs on which the cop wins is closed under the operation of subdividing edges. We also give a complete answer to the question of when the cop has a winning strategy on K 1/m a,b .

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