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).