Integral bases and p-twisted digraphs

A well-known theorem in network flow theory states that for a strongly connected digraph D = (V, A) there exists a set of directed cycles the incidence vectors of which form a basis for the circulation space of D and integrally span the set of integral circulations; that is, every integral circulation can be written as an integral combination of these vectors. In this paper, we extend this result to general digraphs. Following a definition of Hershkowitz and Schneider, we call a digraph p-twisted if each pair of vertices is contained in a closed (undirected) walk with the property that as the walk is traversed there are no more than p changes in the orientations of the arcs. We show that for every p-twisted digraph there exists a set of p-twisted cycles the incidence vectors of which form a basis for the circulation space and integrally span the set of integral circulations. We show that such a set can be computed in O(|||) time.