Localization game on geometric and planar graphs

TL;DR

This paper introduces a pursuit game on graphs inspired by localization problems in cellular networks, analyzing the minimum number of agents needed to locate a moving target across various graph classes and their algorithmic complexities.

Contribution

It defines a new graph invariant related to localization, provides bounds for different graph classes, and establishes complexity and approximation results for the problem.

Findings

Existence of planar graphs with treewidth 2 and unbounded localization number

Localization number bounded by the graph's pathwidth

NP-hardness of the localization problem in diameter-2 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 $\zeta (G)$, 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 $\zeta (G)$. On a positive side, we prove that $\zeta (G)$ is bounded by the pathwidth of $G$. We then show that the algorithmic problem of determining $\zeta (G)$ 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.