Progress on the localization number of a graph

TL;DR

This paper reviews recent advances in understanding the localization number of graphs, focusing on bounds and exact values for specific classes like incidence, polarity, and Kneser graphs, based on a graph searching game.

Contribution

It presents new bounds and exact values for the localization number of various graph classes, expanding the theoretical understanding of this parameter.

Findings

Bounds and exact values for incidence graphs of designs

Results on polarity graphs

Localization number of Kneser graphs

Abstract

We highlight new results on the localization number of a graph, a parameter derived from the localization graph searching game. After introducing the game and providing an overview of existing results, we describe recent results on the localization number. We describe bounds or exact values of the localization number of incidence graphs of designs, polarity graphs, and Kneser graphs.