Some results about a conjecture on identifying codes in complete suns

Consider a graph G = ( V , E ) and, for every vertex v ∈ V , denote by B ( v ) the set { v } ∪ { u : u v ∈ E } . A subset C ⊆ V is an identifying code if the sets B ( v ) ∩ C , v ∈ V , are nonempty and distinct. It is a locating–dominating code if the sets B ( v ) ∩ C , v ∈ V ∖ C , are nonempty and distinct. Let S n be the graph whose vertex set can be partitioned into two sets, U n and V n , whereU n = { u 1 , u 2 , … , u n }induces a clique andV n = { v 1 , 2 , v 2 , 3 , … , v n − 1 , n , v n , 1 }induces an independent set, with edgesv i , i + 1u iandv i , i + 1u i + 1, 1 ≤ i ≤ n ; computations are carried modulo  n . This graph is called a complete sun. We prove the conjecture that the smallest identifying code in  S n has size equal to  n . We also characterize and count all the identifying codes with size  n in  S n . Finally, we determine the sizes of the smallest LD codes in  S n .