Integral bases and \(p\)-twisted digraphs (Q1896335)

A digraph is called \(p\)-twisted if each pair of vertices is a part of a closed walk such that if the walk is traversed once there are at most \(p\) twists, i.e. changes in the orientation of the edges. The circulation space of a directed graph \(D= (V, A)\) with \(v(D)\) weak components is a subspace of dimension \(|A|- |V|+ v(D)\) of the vector space of all real-valued vectors indexed on the arcs \(A\). It is shown 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. This statement extends a result for strongly connected digraphs.