Strong domination number of some operations on a graph

Let $G=(V(G),E(G))$ be a simple graph. A set $D\subseteq V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V(G)\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $\deg(x)\leq \deg(y)$. The strong domination number $\gamma_{st}(G)$ is defined as the minimum cardinality of a strong dominating set.  In this paper, we examine the effects on $\gamma_{st}(G)$ when $G$ is modified by operations on edge (or edges) of $G$.