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 $|N[v]\cap S|\geq 2$ 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 $γ_{\times 2}(G)$, is the cardinality of a smallest double dominating set of $G$. A graph $G$ is said to be double domination edge critical if $γ_{\times 2}(G+e)<γ_{\times 2}(G)$ for any edge $e \notin E$. A double domination edge critical graph $G$ with $γ_{\times 2}(G)=k$ is called $k$-$γ_{\times 2}(G)$-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 $4$-$γ_{\times 2}(G)$-critical graph of odd order with minimum degree at least 4 except a family of graphs.

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