  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 $deg(x)\leq deg(y)$. The strong domination number
$\gamma_{st}(G)$ is defined as the minimum cardinality of a strong dominating
set. In this paper, we study the strong domination number of Haj\'{o}s sum and
vertex-sum of two graphs.
