Characterizing defect n -extendable graphs and (2n+1)(2n+1)-critical graphs

A near perfect matching is a matching saturating all but one vertex in a graph. Let G be a connected graph. If any n independent edges in G are contained in a near perfect matching where n   is a positive integer and n⩽(|V(G)|-2)/2n⩽(|V(G)|-2)/2, then G is said to be defect n-extendable. If deleting any k vertices in G   where k⩽|V(G)|-2k⩽|V(G)|-2, the remaining graph has a perfect matching, then G is a k-critical graph. This paper first shows that the connectivity of defect n-extendable graphs can be any integer. Then the characterizations of defect n  -extendable graphs and (2k+1)(2k+1)-critical graphs using M-alternating paths are presented.