On the number of optimal identifying codes in a twin-free graph

Let G be a simple, undirected graph with vertex set V . For v ∈ V and r ≥ 1, we denote by BG,r (v) the ball of radius r and centre v. A set C ⊆ V is said to be an r-identifying code in G if the sets BG,r (v) ∩ C , v ∈ V , are all nonempty and distinct. A graph G which admits an r-identifying code is called r-twin-free or r-identifiable, and in this case the smallest size of an r-identifying code in G is denoted by γ ID r (G). We study the number of different optimal r-identifying codes C , i.e., such that |C | = γ ID r (G), that a graph G can admit, and try to construct graphs having ‘‘many’’ such codes. © 2014 Elsevier B.V. All rights reserved.