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We introduce the notion of watching systems in graphs, which is a generalization of that of identifying codes. We give some basic properties of watching systems, an upper bound on the minimum size of a watching system, and results on the graphs which achieve this bound; we also study the cases of the paths and cycles, and give complexity results.

Let G be a simple, undirected, connected graph with vertex set V(G) and C V(G) be a set of vertices whose elements are called codewords. For vAV(G) and rX1, let us denote by I r C (v) the set of codewords cAC such that d(v, c)4r, where the distance d(v, c) is defined as the length of a shortest path between v and c. More generally, for A V(G), we define I C r ðAÞ ¼ [ v2A I C r ðvÞ, which is the set of codewords whose minimum distance to an element of A is at most r. If r and l are positive integers, C is said to be an (r, 4l)identifying code if one has I r C (A)6 ¼I r C (A 0 ) whenever A and A 0 are distinct subsets of V(G) with at most l elements. We consider the problem of finding the minimum size of an (r, 4l)-identifying code in a given graph. It is already known that this problem is NP-hard in the class of all graphs when l 5 1 and rX1. We show that it is also NP-hard in the class of planar graphs with maximum degree at most three for all (r, l) with rX1 and lAf1, 2g. This shows, in particular, that the problem of computing the minimum size of an (r, 42)-identifying code in a given graph is NP-hard.