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 $D \subseteq V(G)$ is a \emph{strong defensive alliance} if for every vertex $v \in D$ the condition $\delta_D(v) \geq \delta_{\overline{D}}(v)$ holds. The \emph{strong defensive alliance number} $\alpha(G)$ is defined as the minimum cardinality among all the strong defensive alliances. A \emph{unitary operator} of graphs $\mathcal{O}$ assigns to each graph $G$ a graph $\mathcal{O}(G)$. 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 $\alpha(S(G))$ and $\alpha(R(G))$. 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 $\alpha(\overline{S(G)})$, $\alpha(Q(G))$, $\alpha(\overline{Q(G)})$, $\alpha(T(G))$, and $\alpha(\C{G})$.