We consider a game in which a cop searches for a moving robber on a graph
using distance probes, which is a slight variation on one introduced by Seager.
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 G1/m obtained by
replacing each edge of G by a path of length m, if m⩾n. They
conjectured that this bound was best possible for complete graphs, but the
present authors showed that in fact the cop wins on K1/m if and only if m⩾n/2, for all but a few small values of n. In this paper we extend
this result to general graphs by proving that the cop has a winning strategy on
G1/m provided m⩾n/2 for all but a few small values of n;
this bound is best possible. We also consider replacing the edges of G with
paths of varying lengths.Comment: 13 Page