The minimum degree of minimal 2-extendable claw-free graphs

A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1\le k\le \frac{\mathrm{\mid }V(G)\mathrm{\mid }}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal if the deletion of any edge results in a graph that is not $k$-extendable. In 1994, Plummer proved that every $k$-extendable claw-free graph has minimum degree at least $2k$. Recently, He et al. showed that every minimal 1-extendable graph has minimum degree 2 or 3. In this paper, we prove that the minimum degree of a minimal 2-extendable claw-free graph is either $4$ or $5$.