Acyclic digraphs and local hierarchy theory

The degree of a vertex of a digraph is the number of outgoing edges minus the number of incoming edges. Acyclic digraphs give a model for networks such as citation networks and organizational charts. Motivated by a “local hierarchy theory” developed for this model, we consider the set D ̂ ( δ ) of acyclic digraphs with a specified degree sequence δ . We show that all digraphs in this set can be generated from any one such digraph using just one kind of basic transformation. In the case of degree sequences δ that are minimal in the “Lorenz order”, we investigate the maximum number of edges in an acyclic digraph in D ̂ ( δ ) and show how to construct digraphs in D ̂ ( δ ) with many edges. Determining this maximum number of edges seems to be a very difficult problem.