  A vertex subset $S$ of a graph $G$ is a double dominating set of $G$ if
$|N[v]\cap S|\geq 2$ for each vertex $v$ of $G$, where $N[v]$ is the set of the
vertex $v$ and vertices adjacent to $v$. The double domination number of $G$,
denoted by $\gamma_{\times 2}(G)$, is the cardinality of a smallest double
dominating set of $G$. A graph $G$ is said to be double domination edge
critical if $\gamma_{\times 2}(G+e)<\gamma_{\times 2}(G)$ for any edge $e
\notin E$. A double domination edge critical graph $G$ with $\gamma_{\times
2}(G)=k$ is called $k$-$\gamma_{\times 2}(G)$-critical. A graph $G$ is
$r$-factor-critical if $G-S$ has a perfect matching for each set $S$ of $r$
vertices in $G$. In this paper we show that $G$ is 3-factor-critical if $G$ is
a 3-connected claw-free $4$-$\gamma_{\times 2}(G)$-critical graph of odd order
with minimum degree at least 4 except a family of graphs.

Combinatorics