Edge metric dimension of some graph operations

TL;DR

This paper investigates the edge metric dimension of various graph operations, such as join, lexicographic, and corona products, providing insights into how these operations affect the minimum size of edge metric generators.

Contribution

It introduces the concept of edge metric dimension for graph operations and analyzes its behavior for join, lexicographic, and corona products.

Findings

Edge metric dimension is characterized for join, lexicographic, and corona products.

The paper establishes bounds and exact values for the edge metric dimension in these graph operations.

Results contribute to understanding how graph operations influence metric-based graph parameters.

Abstract

Let $G=(V, E)$ be a connected graph. Given a vertex $v\in V$ and an edge $e=uw\in E$, the distance between $v$ and $e$ is defined as $d_G(e,v)=\min\{d_G(u,v),d_G(w,v)\}$. A nonempty set $S\subset V$ is an edge metric generator for $G$ if for any two edges $e_1,e_2\in E$ there is a vertex $w\in S$ such that $d_G(w,e_1)\ne d_G(w,e_2)$. The minimum cardinality of any edge metric generator for a graph $G$ is the edge metric dimension of $G$. The edge metric dimension of the join, lexicographic and corona product of graphs is studied in this article.