
The talk presents a joint work of Jarosław Grytczuk, Joanna Sokół, Małgorzata ŚleszyńskaNowak. We consider the following metric version of the Cops and Robbers game. Let G be a simple graph and let k≥1 be a fixed integer. In the first round, Cop picks a subset of k vertices B={v_{1},v_{2},…,v_{k}} and then Robber picks a vertex u but keeps it in a secret. Then Cop asks Robber for a vector D_{u}(B)=(d_{1},d_{2},…,d_{k}) whose components d_{i}=d_{G}(u,v_{i}), i=1,2,…,k, are the distances from u to the vertices of B. In the second round, Robber may stay at the vertex u or move to any neighbouring vertex which is kept in a secret. Then Cop picks another k vertices and asks as before for the corresponding distances to the vertex occupied by Robber. And so on in every next round. The game stops when Cop determines exactly the current position of Robber. In that case, she is the winner. Otherwise, Robber is the winner (that is if Cop is not able to localize him in any finite number of rounds). Let ζ(G) denote the least integer k for which Cop has a winning strategy. Notice that this parameter is well defined since the inequality ζ(G)≤V(G) holds obviously. 