On $(n-2)$-Extendable Graphs-II

Abstract

A simple graph $G$ with a perfect matching is said to be \emph{$k$-extendable} if for every set $M$ of $k$ independent edges, there exists a perfect matching in $G$ containing all the edges of $M$. In an earlier paper, we characterized $(n-2)$-extendable graphs on $2n\ge 10$ vertices. In this paper, we complete the characterization by resolving the remaining small cases of $2n=6$ and $8$. In addition, the subclass of $k$-extendable graphs that are “critical” and “minimal” are determined.