Complexity results for identifying codes in planar graphs

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 v ∈ V (G) and r ≥ 1, let us denote by I r (v) the set of codewords c ∈ C such that d(v, c) ≤ r, 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 r (A) = ∪v∈AI C r (v), 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,≤ l)-identifying code if one has I r (A) 6= I C r (A ) whenever A and A are distinct subsets of V (G) with at most l elements. We consider the problem of finding the minimum size of an (r,≤ l)identifying code in a given graph. It is already known that this problem is NP -hard in the class of all graphs when l = 1 and r ≥ 1. 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 r ≥ 1 and l ∈ {1, 2}. This shows, in particular, that the problem of computing the minimum size of an (r,≤ 2)-identifying code in a given graph is NP -hard.