Existence of acyclic matching and Morse complex on transitive digraphs

For any digraph, there exists a transitive closure. The transitive digraph is a discrete geometric object which has a close relationship with simplicial complex. In this paper, based on the discrete Morse theory of simplicial complexes and general chain complexes, we analyze the properties of paths on digraphs and prove the existence of acyclic matching on transitive digraphs, as well as the isomorphism between the homology groups of Morse complex and path homology groups for transitive digraphs.