Strong resolving graphs: the realization and the characterization problems

TL;DR

This paper surveys the current understanding of strong resolving graphs, their properties, and their role in determining the strong metric dimension of graphs, including new results on their characterization and realization.

Contribution

It provides a comprehensive survey of strong resolving graphs and introduces new findings on their realization and characterization problems.

Findings

Derived new results on the properties of strong resolving graphs

Provided characterization criteria for strong resolving graphs

Explored the realization problem for strong resolving graphs

Abstract

The strong resolving graph $G_{SR}$ of a connected graph $G$ was introduced in [Discrete Applied Mathematics 155 (1) (2007) 356--364] as a tool to study the strong metric dimension of $G$. Basically, it was shown that the problem of finding the strong metric dimension of $G$ can be transformed to the problem of finding the vertex cover number of $G_{SR}$. Since then, several articles dealing with this subject have been published. In this paper, we survey the state of knowledge on the strong resolving graph and also derive some new results.