Localization game on geometric and planar graphs

Abstract

The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph G we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the metric dimension of a graph. We provide upper bounds on the related graph invariant ζ(G), defined as the least number of cops needed to localize the robber on a graph G, for several classes of graphs (trees, bipartite graphs, etc.). Our main result is that, surprisingly, there exists planar graphs of treewidth 2 and unbounded ζ(G). On a positive side, we prove that ζ(G) is bounded by the pathwidth of G. We then show that the algorithmic problem of determining ζ(G) is NP-hard in graphs with diameter at most 2. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane.