The existence of a graph whose vertex set can be partitioned into a fixed number of strong domination-critical vertex-sets

Let $\gamma (G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a critical vertex of $G$ if $\gamma (G-v)=\gamma (G)-1$. A graph is called vertex-critical if its every vertex is critical. In this paper, we correspondingly introduce two such definitions: (i) A set $S\subseteq V(G)$ is called a strong critical vertex-set of $G$ if $\gamma (G-S)=\gamma (G)-|S|$; (ii) A graph $G$ is called strong $l$-vertex-set-critical if $V(G)$ can be partitioned into $l$ strong critical vertex-sets of $G$. Therefrom, we give some properties of strong $l$-vertex-set-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 prove that there exists a strong $l$-vertex-set-critical connected graph if and only if $l\notin \{2,3,5\}$.