The k-metric dimension of a graph

TL;DR

This paper introduces the concept of the $k$-metric basis in graphs, generalizing metric bases by requiring vertices to distinguish pairs with at least $k$ elements, and explores conditions and results related to this new measure.

Contribution

It defines the $k$-metric basis and dimension, provides necessary and sufficient conditions for a graph to be $k$-metric dimensional, and presents several new theoretical results.

Findings

Characterization of $k$-metric dimensional graphs

Conditions for the existence of $k$-metric bases

Results on the $k$-metric dimension of graphs

Abstract

As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of any pair of different vertices of $G$ are distinguished by at least $k$ elements of $S$, i.e., for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. A connected graph $G$ is $k$-metric dimensional if $k$ is the largest integer such that there exists a $k$-metric basis for $G$. We give a necessary and sufficient condition for a graph to be $k$-metric dimensional and we obtain several results on the $k$-metric dimension.