Separation-network synthesis (SNS) is carried out by a heuristic, algorithmic or combined method. For any separation-network, the algorithmic method may yield the optimal structure provided that a valid mathematical model of the network structure is available. In general, a mathematical model of the network structure is constructed by imposing constraints on some structural properties of optimal separation-networks, e.g., exclusion of redundant separations. The fundamental structural properties of separation-networks have been explored in the current work. The validity of the mathematical model, specifically with respect to redundancy, has been systematically studied. It is unequivocally demonstrated that redundant separators may be contained in an optimal separation-network under certain circumstances. Two definite classes of problems of separation-network synthesis have been parametrically studied extensively for illustration.