The localization number and metric dimension of graphs of diameter 2

TL;DR

This paper investigates the localization number and metric dimension of diameter 2 graphs, including Kneser, Moore, and polarity graphs, providing bounds and exact values that advance understanding of these parameters.

Contribution

It offers new bounds and exact values for the localization number and metric dimension of specific diameter 2 graphs, improving previous results especially for Kneser graphs.

Findings

Bounds for localization number and metric dimension of Kneser graphs are established.

Exact metric dimension values are obtained for infinitely many Kneser graphs.

Bounds are also provided for Moore and polarity graphs.

Abstract

We consider the localization number and metric dimension of certain graphs of diameter $2$, focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with diameter $2$, we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter $2$ and polarity graphs.