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Let $\gamma(G)$ denote the domination number of a graph $G$. A vertex $v\inV(G)$ is called a \emph{critical vertex} of $G$ if $\gamma(G-v)=\gamma(G)-1$. Agraph is called \emph{vertex-critical} if every vertex of it is critical. Inthis 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 criticalvertex-sets of $G$. Whereafter, we give some properties of strong$l$-vertex-sets-critical graphs by extending the previous results ofvertex-critical graphs. As the core work, we study on the existence of thisclass of graphs and obtain that there exists a strong $l$-vertex-sets-criticalconnected graph if and only if $l\notin\{2,3,5\}$.

The Existence of Graph whose Vertex Set Can be Partitioned into a Fixed Number of Domination Strong Critical Vertex-sets