  For a graph $G=(V,E)$ of order $n$, a Roman $\{2\}$-dominating function
$f:V\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V$ with
$f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$, or $v$ is
adjacent to least two vertices assigned $1$ under $f$. In this paper, we
classify all graphs with Roman $\{2\}$-domination number belonging to the set
$\{2,3,4,n-2,n-1,n\}$. Furthermore, we obtain some results about Roman
$\{2\}$-domination number of some graph operations.
