Infinite graphs with finite metric dimension

TL;DR

This paper investigates the conditions under which infinite graphs have finite metric dimensions, providing characterizations based on graph ends and vertex degrees.

Contribution

It offers new characterizations of infinite graphs with finite strong and weak metric dimensions based on their end structure and vertex degree properties.

Findings

Graphs with more than one end have infinite strong dimension.

Finite weak dimension occurs iff finitely many degree-three vertices in graphs with cycles.

Finite strong dimension occurs iff the graph has one end and finitely many degree-three vertices.

Abstract

We study the metric dimension (strong and weak) of infinite graphs. In particular, our main interest is characterizing infinite graphs with finite dimension. Our main results: (1) graphs with more than one end have infinite strong dimension; (2) for graphs with a finite number of cycles, the weak dimension is finite if and only if the graph has finitely many vertices of degree three, and the strong dimension is finite if and only if the graph has one end and finitely many vertices of degree three.