In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of paths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.

In this paper, we give a necessary and sufficient condition that discrete Morse functions on a digraph can be extended to be Morse functions on its transitive closure, from this we can extend the Morse theory to digraphs by using quasi-isomorphism between path complex and discrete Morse complex, we also prove a general sufficient condition for digraphs that the Morse functions satisfying this necessary and sufficient condition.

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We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy invariance of homologies of digraphs and the relation between the fundamental group of the digraph and its first homology group. The category of (undirected) graphs can be identified by a natural way with a full subcategory of digraphs. Thus we obtain also consistent homology and homotopy theories for graphs.