
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},_{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 localise 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. The aim of the talk is to present results concerning 2trees, outerplanar graphs and planar graphs. This is a joint work with Przemysław Gordinowicz, Jarosław Grytczuk, Nicolas Nisse, Joanna Sokół, and Małgorzata ŚleszyńskaNowak.
Bartłomiej Bosek, Przemysław Gordinowicz, Jarosław Grytczuk, Nicolas Nisse, Joanna Sokół, Małgorzata ŚleszyńskaNowak. Centroidal localization game. arXiv, pages 115, 2017.
Bartłomiej Bosek, Przemysław Gordinowicz, Jarosław Grytczuk, Nicolas Nisse, Joanna Sokół, Małgorzata ŚleszyńskaNowak. Localization game on geometric and planar graphs. arXiv, pages 115, 2017. 