A note on the complexity of k-Metric Dimension

01/28/2021

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by   Yannick Schmitz, et al.

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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 w_1,…,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.