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) boolean AND C, v is an element of V (respectively, v is an element of V \ C), 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. (C) 2002 Elsevier Science B.V. All rights reserved.