  Let $\gamma(G)$ denote the domination number of a graph $G$. A vertex $v\in
V(G)$ is called a \emph{critical vertex} of $G$ if $\gamma(G-v)=\gamma(G)-1$. A
graph is called \emph{vertex-critical} if every vertex of it is critical. In
this paper, we correspondingly introduce two such definitions: (i) a set
$S\subseteq V(G)$ is called a \emph{strong critical vertex-set} of $G$ if
$\gamma(G-S)=\gamma(G)-|S|$; (ii) a graph $G$ is called \emph{strong
$l$-vertex-sets-critical} if $V(G)$ can be partitioned into $l$ strong critical
vertex-sets of $G$. Whereafter, we give some properties of strong
$l$-vertex-sets-critical graphs by extending the previous results of
vertex-critical graphs. As the core work, we study on the existence of this
class of graphs and obtain that there exists a strong $l$-vertex-sets-critical
connected graph if and only if $l\notin\{2,3,5\}$.
