The matching extension problem in general graphs is co-NP-complete

A simple connected graph G with 2n vertices is said to be k-extendable for an integer k with 0<k<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<k<n$$\end{document} if G contains a perfect matching and every matching of cardinality k in G is a subset of some perfect matching. Lakhal and Litzler (Inf Process Lett 65(1):11–16, 1998) discovered a polynomial algorithm that decides whether a bipartite graph is k-extendable. For general graphs, however, it has been an open problem whether there exists a polynomial algorithm. The new result presented in this paper is that the extendability problem is co-NP-complete.