A note on the complexity of k-Metric Dimension

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 $w_1,\ldots,w_k \in R$ 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 $k \geq 2$.