Localization Game for Random Graphs

TL;DR

This paper studies the localization game on graphs, focusing on the localization number, and provides improved bounds for dense random graphs as well as extending results to sparse graphs.

Contribution

It offers new bounds and asymptotic behavior analysis for the localization number in both dense and sparse random graphs.

Findings

Improved bounds for the localization number in dense random graphs.

Asymptotic behavior of the localization number established.

Extension of results to sparse graphs.

Abstract

We consider the localization game played on graphs in which a cop tries to determine the exact location of an invisible robber by exploiting distance probes. The corresponding graph parameter $\zeta(G)$ for a given graph $G$ is called the localization number. In this paper, we improve the bounds for dense random graphs determining an asymptotic behaviour of $\zeta(G)$. Moreover, we extend the argument to sparse graphs.