The Robber Locating game

Abstract

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 .

Locating a robber with multiple probes

“…that K 1/m n is locatable if and only if m ≥ n. The present authors showed that in fact K 1/m n is locatable if and only if m ≥ n/2, for every n ≥ 11 [12], and then subsequently that the same improvement on the upper bound may be obtained in general: provided |V (G)| ≥ 23, G 1/m is locatable whenever m ≥ |V (G)|/2 [13]. This bound is best possible, since K 1/m n is not locatable if m = (n − 1)/2, and some lower bound on |V (G)| is required for it to hold, since K 1/5 10 is not locatable [12]. These results fundamentally depend on taking equal-length subdivisions, and do not imply any results for unequal subdivisions, since subdividing a single edge of a locatable graph can result in a non-locatable graph (as observed independently by Seager [26] and in [12]); however, the present authors showed that an unequal subdivision is also locatable provided every edge is subdivided into a path of length at least 2|V (G)| [13].…”

“…In most graphs this bound is simply |V (G)|, and they conjectured that this was best possible for complete graphs, i.e. that K 1/m n is locatable if and only if m ≥ n. The present authors showed that in fact K 1/m n is locatable if and only if m ≥ n/2, for every n ≥ 11 [12], and then subsequently that the same improvement on the upper bound may be obtained in general: provided |V (G)| ≥ 23, G 1/m is locatable whenever m ≥ |V (G)|/2 [13]. This bound is best possible, since K 1/m n is not locatable if m = (n − 1)/2, and some lower bound on |V (G)| is required for it to hold, since K 1/5 10 is not locatable [12].…”

“…In this paper we consider the Robber Locating game, introduced in a slightly different form by Seager [24] and further studied by Carragher, Choi, Delcourt, Erickson and West [4], as well as by the current authors [12,13]. This combines features of the other games mentioned above.…”

“…This bound is best possible, since K 1/m n is not locatable if m = (n − 1)/2, and some lower bound on |V (G)| is required for it to hold, since K 1/5 10 is not locatable [12]. These results fundamentally depend on taking equal-length subdivisions, and do not imply any results for unequal subdivisions, since subdividing a single edge of a locatable graph can result in a non-locatable graph (as observed independently by Seager [26] and in [12]); however, the present authors showed that an unequal subdivision is also locatable provided every edge is subdivided into a path of length at least 2|V (G)| [13].…”

Subdivisions in the Robber Locating game

“…In most graphs this bound is simply |V (G)|, and they conjectured that this was best possible for complete graphs, i.e. that K 1/m n is locatable if and only if m n. The present authors [8] showed that in fact K 1/m n is locatable if and only if m n/2, for every n 11. In this paper we show that the same improvement may be obtained in general: provided |V (G)| 23, G 1/m is locatable whenever m |V (G)|/2.…”