  In this paper, we consider 2-cographs, a natural generalization of the
well-known class of cographs. We show that, as with cographs, 2-cographs can be
recursively defined. However, unlike cographs, 2-cographs are closed under
induced minors. We consider the class of non-2-cographs for which every proper
induced minor is a 2-cograph and show that this class is infinite. Our main
result finds the finitely many members of this class whose complements are also
induced-minor-minimal non-2-cographs.
