Two vertices u, v ∈ 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 ⊆ V of vertices is a k-resolving set for G if for each pair of vertices u, v ∈ V there are at least k distinct vertices w1, . . . , w k ∈ 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 ≥ 2.

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Let G be a strongly connected directed graph and u, v, w ∈ V (G) be three vertices. Then w strongly resolves u to v if there is a shortest u-w-path containing v or a shortest w-v-path containing u. A set R ⊆ V (G) of vertices is a strong resolving set for a directed graph G if for every pair of vertices u, v ∈ V (G) there is at least one vertex in R that strongly resolves u to v and at least one vertex in R that strongly resolves v to u. The distances of the vertices of G to and from the vertices of a strong resolving set R uniquely define the connectivity structure of the graph. The Strong Metric Dimension of a directed graph G is the size of a smallest strong resolving set for G. The decision problem Strong Metric Dimension is the question whether G has a strong resolving set of size at most r, for a given directed graph G and a given number r. In this paper we study undirected and directed co-graphs and introduce linear time algorithms for Strong Metric Dimension. These algorithms can also compute strong resolving sets for co-graphs in linear time.