Abstract This paper focuses on the design of minimum cost networks satisfying two technical constraints. First, the connected components should be unicyclic. Second, some given special nodes must belong to cycles. This problem is a generalization of the perfect binary 2-matching problem. It turns out that the problem is easy to solve since it can be seen as a b-matching in an appropriate extended graph. We also present a partial description of the convex hull of the incidence vectors of these Steiner networks. Polynomial time separation algorithms are described. One of them is a generalization of the Padberg-Rao algorithm to separate blossom inequalities.