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 \emph{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.