Metric dimension of growing infinite graphs

TL;DR

This paper studies how adding edges to infinite graphs affects their metric dimension, revealing that it can oscillate infinitely often between finite and infinite, but small edge modifications cannot cause such drastic changes.

Contribution

It demonstrates that the metric dimension of infinite graphs can oscillate infinitely often with edge additions, but small changes cannot alter its finiteness.

Findings

Metric dimension can change infinitely often between finite and infinite

Small edge modifications do not cause drastic changes in metric dimension

Constructs a sequence of graphs with oscillating metric dimension

Abstract

We investigate how the metric dimension of infinite graphs change when we add edges to the graph. Our two main results: (1) there exists a growing sequence of graphs (under the subgraph relation, but without adding vertices) for which the metric dimension changes between finite and infinite infinitely many times; (2) finite changes in the edge set can not change the metric dimension from finite to infinite or vice versa.