Edge Resolvability for Circular Ladder of Heptagons

TL;DR

This paper investigates the edge and vertex metric dimensions of a heptagonal circular ladder and related convex polytope graphs, establishing their equality and independence of minimal generators.

Contribution

It determines the edge metric dimension of the heptagonal circular ladder and introduces a new convex polytope graph family with its metric dimension, highlighting their independence.

Findings

Edge metric dimension of heptagonal circular ladder is three.

Edge and vertex metric dimensions are equal for these graphs.

Minimal generators are independent across the studied graph families.

Abstract

A set $\mathbb{Y}$ of elements (vertices or edges) in space is said to be a $generator$ of a metric space if each element of the space is recognized by its distances from the elements of $\mathbb{Y}$, uniquely. The generator with minimum cardinality is known as the $basis$ of the metric space, and this cardinality is the $dimension$ of the given space. In this article, we further discuss these notions with respect to a heptagonal circular ladder. We show that for a heptagonal circular ladder $\Gamma_{n}$, the edge metric dimension is three and find that it equals its metric dimension. We also introduce a new family of the convex polytope graph (denoted by $\Delta_{n}$) from a heptagonal circular ladder and find its metric dimension. Furthermore, we prove that the minimum generator (metric and edge metric) are independent for all of these families of the convex polytopes.