A note on $k$-metric dimensional graphs

TL;DR

This paper investigates bounds on the maximum integer k for which a graph has a k-metric generator, extending the understanding of k-metric dimensional graphs and their properties.

Contribution

It provides new bounds and insights into the k-metric dimensionality of graphs, a concept related to vertex distinguishability.

Findings

Derived bounds for k-metric dimensional graphs

Characterized graphs based on their k-metric properties

Extended previous concepts of metric dimension

Abstract

Given a graph $G = (V,E)$, a set $S \subset V$ is called a $k$-\emph{metric generator} for $G$ if any pair of different vertices of $G$ is distinguished by at least $k$ elements of $S$. A graph is $k$-\emph{metric dimensional} if $k$ is the largest integer such that there exists a $k$-metric generator for $G$. This paper studies some bounds on the number $k$ for which a graph is $k$-metric dimensional.