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arXiv · 1401.0342

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

Ismael G. Yero·Alejandro Estrada-Moreno·Juan A. Rodriguez-Velazquez
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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.

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Ismael G. Yero, Alejandro Estrada-Moreno, Juan A. Rodriguez-Velazquez. 2014-01-01. On the complexity of computing the $k$-metric dimension of graphs. https://arxiv.org/abs/1401.0342

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