On the Metric Dimension of Signed Graphs

TL;DR

This paper extends the concept of metric dimension to signed graphs, analyzing its properties and invariance under negation, with specific focus on well-known classes including signed trees.

Contribution

It introduces the metric dimension for signed graphs and proves its invariance under negation, expanding the understanding of resolving sets in signed graph theory.

Findings

Metric dimension is invariant under negation in signed graphs

Analyzed metric dimension for classes including signed trees

Extended concepts from unsigned to signed graphs

Abstract

A signed graph $\Sigma$ is a pair $(G,\sigma)$, where $G=(V,E)$ is the underlying graph in which each edge is assigned $+1$ or $-1$ by the signature function $\sigma:E\rightarrow\{-1,+1\}$. In this paper, we extend the extensively applied concepts of metric dimension and resolving sets for unsigned graphs to signed graphs. We analyze the metric dimension of some well known classes of signed graphs including a special case of signed trees. Among other things, we establish that the metric dimension of a signed graph is invariant under negation.