Characterization of the induced matching extendable graphs with 2n vertices and 3n edges

A graph G is induced matching extendable or IM-extendable if every induced matching of G is contained in a perfect matching of G. In 1998, Yuan proved that a connected IM-extendable graph on 2n vertices has at least 3n−2 edges, and that the only IM-extendable graph with 2n vertices and 3n−2 edges is T×K2 , where T is an arbitrary tree on n vertices. In 2005, Zhou and Yuan proved that the only IM-extendable graph with 2n≥6 vertices and 3n−1 edges is T×K2+e, where T is an arbitrary tree on n vertices and e is an edge connecting two vertices that lie in different copies of T and have distance 3 between them in T×K2. In this paper, we introduced the definition of Q-joint graph and characterized the connected IM-extendable graphs with 2n≥4 vertices and 3n edges.