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 γrID(G). We study the number of different optimal r-identifying codes C, i.e., such that |C|=γrID(G), that a graph G can admit, and try to construct graphs having "many" such codes.