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

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 \(2n \geq 10\) 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.