Supermodularity in Unweighted Graph Optimization II: Matroidal Term Rank Augmentation

TL;DR

This paper extends Ryser's max term rank formula to include matroidal models and augmentation, unifying these approaches into a comprehensive framework for bipartite graph degree sequence characterization.

Contribution

It introduces a unified framework combining matroidal models and augmentation for bipartite graph degree sequences, expanding on previous generalizations.

Findings

Develops a matroidal model for degree sequence characterization.

Provides an augmentation version of Ryser's theorem.

Integrates the two approaches into a single framework.

Abstract

Ryser's max term rank formula with graph theoretic terminology is equivalent to a characterization of degree sequences of simple bipartite graphs with matching number at least $\ell$. In a previous paper by the authors, a generalization was developed for the case when the degrees are constrained by upper and lower bounds. Here two other extensions of Ryser's theorem are discussed. The first one is a matroidal model, while the second one settles the augmentation version. In fact, the two directions shall be integrated into one single framework.