Algorithms for realizing degree sequences of directed graphs

The Havel-Hakimi algorithm for constructing realizations of degree sequences for undirected graphs has been used extensively in the literature. A result by Kleitman and Wang extends the Havel-Hakimi algorithm to degree sequences for directed graphs. In this paper we go a step further and describe a modification of Kleitman and Wang’s algorithm that is a more natural extension of Havel-Hakimi’s algorithm, in the sense that our extension is equivalent to Havel-Hakimi’s algorithm when the degree sequence is Eulerian and has an even degree sum. We identify special degree sequences, called ~ C3-anchored, that are ill-defined for the algorithm and force a particular local structure on all directed graph realizations. We give structural characterizations of these directed graphs, as well as degree sequence characterizations for the ~ C3-anchored set. This allows us to identify the ill-defined sequences, leading to a well-defined algorithm. We end with an application to realizing Eulerian degree sequences with an odd degree sum.