In a network stability problem, the aim is to find stable configurations for a given network of boolean gates. For general networks, the problem is known to be computationally hard. Mayr and Subramanian introduced an interesting class of networks by imposing fanout restrictions at each gate, and showed that network stability on this class of networks is still sufficiently rich to express as special cases stable marriage and stable roommate problems. In this paper we study the sequential and parallel complexity of network stability on networks with restricted fanout. Our approach builds on structural properties of these networks and exposes close ties with the theory of retracts and isometric embeddings for product graphs. This structure then gives new efficient algorithms for questions of representation, enumeration, and optimization in stable matching. © 1992.