On weak metric dimension of digraphs

TL;DR

This paper introduces the concept of weak metric dimension for strongly connected digraphs, establishing bounds, characterizing extremal cases, and classifying specific classes of digraphs based on their weak metric dimension.

Contribution

It defines weak metric dimension for digraphs, derives bounds related to diameter and arc count, and classifies digraphs with specific weak metric dimensions.

Findings

Bounds for the number of arcs based on diameter and weak metric dimension

Characterization of digraphs with extremal arc counts

Classification of vertex-transitive digraphs with weak metric dimension 1

Abstract

Using the two way distance, we introduce the concepts of weak metric dimension of a strongly connected digraph $\Gamma$. We first establish lower and upper bounds for the number of arcs in $\Gamma$ by using the diameter and weak metric dimension of $\Gamma$, and characterize all digraphs attaining the lower or upper bound. Then we study a digraph with weak metric dimension $1$ and classify all vertex-transitive digraphs having weak metric dimension $1$. Finally, all digraphs of order $n$ with weak metric dimension $n-1$ or $n-2$ are determined.