Maximal IM-unextendable graphs

A graph G is maximal IM-unextendable if G is not induced matching extendable and, for every two nonadjacent vertices x and y, G+xy is induced matching extendable. We show in this paper that a graph G is maximal IM-unextendable if and only if G is isomorphic to Mr∨(Ks∨(Kn1∪Kn2∪⋯∪Knt)), where Mr is an induced matching of size r, r⩾1, t=s+2, and each ni is odd.