A note on the complexity of k-Metric Dimension

Abstract

Two vertices $u,v\in V$ of an undirected connected graph $G=(V,E)$ are resolved by a vertex $w$ if the distance between $u$ and $w$ and the distance between $v$ and $w$ are different. A set $R\subseteq V$ of vertices is a $k$-resolving set for $G$ if for each pair of vertices $u,v\in V$ there are at least $k$ distinct vertices $w_1,\dots ,w_k\in R$ such that each of them resolves $u$ and $v$. The $k$-Metric Dimension of $G$ is the size of a smallest $k$-resolving set for $G$. The decision problem $k$-Metric Dimension is the question whether G has a $k$-resolving set of size at most $r$, for a given graph $G$ and a given number $r$. In this paper, we proof the NP-completeness of $k$-Metric Dimension for bipartite graphs and each $k\ge 2$.