Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard

Let G=(V,E) be an undirected graph and C a subset of vertices. If the sets B_r(v)∩C, vV (respectively, vVC), are all nonempty and different, where B_r(v) denotes the set of all points within distance r from v, we call C an r-identifying code (respectively, an r-locating-dominating code). We prove that, given a graph G and an integer k, the decision problem of the existence of an r-identifying code, or of an r-locating-dominating code, of size at most k in G, is NP-complete for any r.