On the complexity of computing the $k$-metric dimension of graphs

TL;DR

This paper investigates the computational complexity of determining the $k$-metric dimension of graphs, proving NP-Completeness in general but providing efficient solutions for trees.

Contribution

It establishes the NP-Completeness of computing the $k$-metric dimension and offers a linear-time algorithm for trees, advancing understanding of this graph parameter.

Findings

NP-Completeness of the general problem

Linear-time solution for trees

Insights into the complexity of $k$-metric dimension

Abstract

Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is a $k$-metric generator for $G$ if for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,...,w_k\in S$ 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 $G$. We study some problems regarding the complexity of some $k$-metric dimension problems. For instance, we show that the problem of computing the $k$-metric dimension of graphs is $NP$-Complete. However, the problem is solved in linear time for the particular case of trees.