On Distance and Strong Metric Dimension of the Modular Product

TL;DR

This paper investigates the distance and strong metric dimension of the modular product of graphs, providing formulas and characterizations that advance understanding of graph metric properties in complex graph constructions.

Contribution

It derives the distance formula for the modular product and characterizes the strong resolving graph, enabling computation of the strong metric dimension for various graph families.

Findings

Derived the distance formula for the modular product

Characterized all edges of the strong resolving graph of the modular product

Computed the strong metric dimension for several infinite graph families

Abstract

The modular product $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g',h')$ of $G\diamond H$ are adjacent if $g=g'$ and $hh'\in E(H)$, or $gg'\in E(G)$ and $h=h'$, or $gg'\in E(G)$ and $hh'\in E(H)$, or (for $g\neq g'$ and $h\neq h'$) $gg'\notin E(G)$ and $hh'\notin E(H)$. We derive the distance formula for the modular product and then describe all edges of the strong resolving graph of $G\diamond H$. This is then used to obtain the strong metric dimension of the modular product on several, infinite families of graphs.