  For many types of graphs, criteria have been discovered that give necessary
and sufficient conditions for an integer sequence to be the degree sequence of
such a graph. These criteria tend to take the form of a set of inequalities,
and in the case of the Erd\H{o}s-Gallai criterion (for simple undirected
graphs) and the Gale-Ryser criterion (for bipartite graphs), it has been shown
that the number of inequalities that must be checked can be reduced
significantly. We show that similar reductions hold for the corresponding
criteria for many other types of graphs, including bipartite r-multigraphs,
bipartite graphs with structural edges, directed graphs, r-multigraphs, and
tournaments. We also prove a reduction for imbalance sequences.
