Strong defensive alliances on graph operators

If G = (V(G), E(G)) is a simple connected graph with vertex set V(G) and edge set E(G) , we say that a subset D ⊆ V(G) is a strong defensive alliance if for every vertex v ∈ D the condition δ D ( v ) ≥ δ D ̅ ( v ) holds. The strong defensive alliance number α(G) is defined as the minimum cardinality among all the strong defensive alliances. A unitary operator of graphs 𝒪 assigns to each graph G a graph 𝒪( G ). A few examples of unitary operators of graphs are: Subdivision S(G), R(G) , Middle Q(G) , Total T(G) , and Central 𝒞(G). In this paper we determine the exact values of α (S(G)) and α (R(G)) . We also characterize the graphs G for which the number of strong defensive alliances is 1, 2, or 3 in Q(G) and T(G) . We also we give tight bounds for α (S̅(G̅)) , α (Q(G)) , α (Q̅(G̅)) , α(T(G)) , and α(𝒞(G)) .