On Bases of the Cycle and Cut Spaces in Digraphs

Abstract

We find a maximal number of directed circuits (directed cocircuits) in a base of a cycle (cut) space of a digraph. We show that this space has a base composed of directed circuits (directed cocircuits) if and only if the digraph is totally cyclic (acyclic). Furthermore, this basis can be considered as an ordered set so that each element of the basis has an arc not contained in the previous elements.