Extremal solutions for Network Flow with Differential Constraints -- A Generalization of Spanning Trees

In network flow problems, there is a well-known one-to-one relationship between extreme points of the feasibility region and trees in the associated undirected graph. The same is true for the dual differential problem. In this paper, we study problems where the constraints of both problems appear simultaneously, a variant which is motivated by an application in the expansion planning of energy networks. We show that all extreme points still directly correspond to graph-theoretical structures in the underlying network. The reverse is generally also true in all but certain exceptional cases. We furthermore characterize graphs in which these exceptional cases never occur and present additional criteria for when those cases do not occur due to parameter values.