  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 can be made
equivalent to Havel-Hakimi's algorithm when the degree sequence has equal in
and out degrees and an even degree sum. We identify special degree sequences,
called directed 3-cycle 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 realizations, as well as
characterizations of the ill-defined degree sequences, leading to a
well-defined algorithm.
