Complexity results for identifying codes in planar graphs

Abstract

AbstractLet 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 r1, let us denote by Ir𝒞(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 , 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 Ir𝒞(A)≠Ir𝒞(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 r1. 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 r1 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.