# Local metric dimension of graphs: generalized hierarchical products and some applications

## TL;DR

This paper investigates the local metric dimension of graphs, especially generalized hierarchical products, providing bounds, exact values, and applications in chemistry and delivery service coding.

## Contribution

It establishes bounds and exact values for the local metric dimension of generalized hierarchical product graphs and applies these results to chemistry and delivery service models.

## Key findings

- Sharp bounds on local metric dimension of generalized hierarchical products
- Determination of local metric dimension for several important graphs
- Application of local metric dimension to coding in delivery services

## Abstract

Let $G$ be a graph and $S\subseteq V(G)$. If every two adjacent vertices of $G$ have different metric $S$-representations, then $S$ is a local metric generator for $G$. A local metric generator of smallest order is a local metric basis for $G$, its order is the local metric dimension of $G$. Lower and upper bounds on the local metric dimension of the generalized hierarchical product are proved and demonstrated to be sharp. The results are applied to determine or bound the dimension of several graphs of importance in mathematical chemistry. Using the dimension, a new model for assigning codes to customers in delivery services is proposed.