Maximal flow through a processing network with the source as the only processing node

For the maximal flow problem in a pure network there is a simple criterion for optimality: "the flow is maximal if and only if there is no flow augmenting path from source to sink". A similar optimality criterion will be stated for the problem of maximal flow in a processing network in which the source is the only processing node. It will be shown that a maximal flow can be determined by considering a sequence of not more than !p(p-I) + I pure maximal flow problems, in which p is the number of arcs incident to the source