Strong domination number of Hajós sum and vertex-sum of two graphs

Let $G=(V,E)$ be a simple graph. A set $D\subseteq V$ is a strong dominating set of $G$, if for every vertex $x\in V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\le deg(y)$. The strong domination number ${\gamma }_{st}(G)$ is defined as the minimum cardinality of a strong dominating set. In this paper, we study the strong domination number of Hajós sum and vertex-sum of two graphs.