3-Factor-criticality in double domination edge critical graphs

A vertex subset $S$ of a graph $G$ is a double dominating set of $G$ if $\mathrm{\mid }N[v]\cap S\mathrm{\mid }\ge 2$ 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 $e \notin E$\mathrm{.}Adoubledominationedgecriticalgraph$G$with$\gamma_{\times 2}(G)=k$iscalled$k$-$\gamma_{\times 2}(G)$-critical\mathrm{.}Agraph$G$is$r$-factor-criticalif$G-S$hasaperfectmatchingforeachset$S$of$r$verticesin$G$\mathrm{.}Inthispaperweshowthat$G$is3-factor-criticalif$G$ 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.