A note on the localization number of random graphs: diameter two case

TL;DR

This paper investigates the localization game on dense random graphs with diameter two, establishing probabilistic bounds on the size of vertex sets needed for a cop to locate a robber.

Contribution

It provides new probabilistic bounds on the minimum size of vertex sets required for localization in dense random graphs with diameter two.

Findings

High probability bounds for set sizes guaranteeing localization

Probabilistic analysis of the localization game on dense graphs

Insights into the localization number in diameter-two random graphs

Abstract

We study the localization game on dense random graphs. In this game, a {\em cop} $x$ tries to locate a {\em robber} $y$ by asking for the graph distance of $y$ from every vertex in a sequence of sets $W_1,W_2,\ldots,W_\ell$. We prove high probability upper and lower bounds for the minimum size of each $W_i$ that will guarantee that $x$ will be able to locate $y$.