This representation is essentially a **chain decomposition** of a digraph: a partitioning of the vertices into a set of *chains* (directed paths), which a description in each path of which vertices which precede it in the *other* paths. 

The following reference [[Esko Nuutila's webpage for his doctoral thesis](] is perhaps not the best reference — either in terms of being current, or in terms of the document format (it's an old version of postscript) — but it will certainly describe the application of chain decompositions to solving the transitive closure of digraphs. (Note that any digraph which you can represent with your representation will be transitively closed, so the relationship between your representation and chain decompositions is indeed quite strong.) I will look on Monday to find a better reference, if no-one else suggests a better one.