On the metric dimension, the upper dimension and the resolving number of graphs

TL;DR

This paper investigates three graph parameters—metric dimension, upper dimension, and resolving number—providing characterizations, confirming conjectures, and establishing limitations on their possible values across graph families.

Contribution

It characterizes graphs with equal metric dimension and resolving number, confirms a conjecture about metric and upper dimension realization, and shows certain resolving numbers are not achievable.

Findings

Graphs with equal metric dimension and resolving number are characterized.

The conjecture on metric and upper dimension realization is affirmed.

No integer greater than or equal to 4 can be the resolving number of an infinite graph family.

Abstract

This paper deals with three resolving parameters: the metric dimension, the upper dimension and the resolving number. We first answer a question raised by Chartrand and Zhang asking for a characterization of the graphs with equal metric dimension and resolving number. We also solve in the affirmative a conjecture posed by Chartrand, Poisson and Zhang about the realization of the metric dimension and the upper dimension. Finally we prove that no integer $a\geq 4$ is realizable as the resolving number of an infinite family of graphs.