Reduced Criteria for Degree Sequences

TL;DR

This paper extends the reduction of inequality criteria for recognizing degree sequences to various graph types, simplifying the verification process for complex graph classes.

Contribution

It generalizes the reduction techniques for degree sequence criteria to multiple graph types, including bipartite r-multigraphs and tournaments.

Findings

Reductions apply to bipartite r-multigraphs

Reductions apply to directed graphs and tournaments

Simplifies checking degree sequence conditions

Abstract

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.