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