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 Ir
𝒞(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
Ir𝒞(A)=∪v∈AIr𝒞(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
𝒞(A)≠Ir
𝒞(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.
