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

Let GG be a simple, undirected graph with vertex set  VV. For v∈Vv∈V and r≥1r≥1, we denote by BG,r(v)BG,r(v) the ball of radius  rr and centre  vv. A set C⊆VC⊆V is said to be an rr-identifying code   in  GG if the sets BG,r(v)∩CBG,r(v)∩C, v∈Vv∈V, are all nonempty and distinct. A graph GG which admits an rr-identifying code is called rr-twin-free   or rr-identifiable  , and in this case the smallest size of an rr-identifying code in  GG is denoted by  γrID(G).We study the number of different optimal rr-identifying codes CC, i.e., such that |C|=γrID(G), that a graph GG can admit, and try to construct graphs having “many” such codes.