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 : uv ∈ 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, where U_n = {u₁, u₂, . . . , u_n} induces a clique and V_n= {v_1,2, v_2,3, . . . , v_n−1,n, v_n,1 } induces an independent set, with edges v_i,i+1u_i and v_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.