The Dulmage-Mendelsohn Decomposition for $b$-Matchings

TL;DR

This paper extends the classical Dulmage-Mendelsohn decomposition to $b$-matchings, providing a structural understanding of maximum $b$-matchings and $b$-verifying sets in bipartite graphs.

Contribution

It develops the theory and properties of the Dulmage-Mendelsohn decomposition specifically for $b$-matchings, generalizing the classical case.

Findings

Established the structure of maximum $b$-matchings.

Characterized the family of $b$-verifying sets.

Extended classical decomposition to a broader setting.

Abstract

We establish the theory of the Dulmage-Mendelsohn decomposition for $b$-matchings. The original Dulmage-Mendelsohn decomposition is a classical canonical decomposition of bipartite graphs, which describes the structures of the maximum $1$-matchings and the dual optimizers, i.e., the minimum vertex covers. In this paper, we develop analogical properties, and thus obtain the structure of the maximum $b$-matchings and characterizes the family of $b$-verifying set.