Localization game on geometric and planar graphs
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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.
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