Nonbipartite Dulmage-Mendelsohn Decomposition for Berge Duality

TL;DR

This paper introduces a new canonical decomposition for arbitrary graphs that generalizes the Dulmage-Mendelsohn decomposition, enabling a unified understanding of canonical decompositions and characterizing maximal barriers in general graphs.

Contribution

It extends the Dulmage-Mendelsohn decomposition to nonbipartite graphs using basilica decomposition, unifying various canonical decompositions and characterizing maximal barriers.

Findings

A new canonical decomposition for arbitrary graphs is established.

The decomposition unifies existing canonical decompositions.

A characterization of maximal barriers in general graphs is provided.

Abstract

The Dulmage-Mendelsohn decomposition is a classical canonical decomposition in matching theory applicable for bipartite graphs, and is famous not only for its application in the field of matrix computation, but also for providing a prototypal structure in matroidal optimization theory. The Dulmage-Mendelsohn decomposition is stated and proved using the two color classes, and therefore generalizing this decomposition for nonbipartite graphs has been a difficult task. In this paper, we obtain a new canonical decomposition that is a generalization of the Dulmage-Mendelsohn decomposition for arbitrary graphs, using a recently introduced tool in matching theory, the basilica decomposition. Our result enables us to understand all known canonical decompositions in a unified way. Furthermore, we apply our result to derive a new theorem regarding barriers. The duality theorem for the maximum matching problem is the celebrated Berge formula, in which dual optimizers are known as barriers. Several results regarding maximal barriers have been derived by known canonical decompositions, however no characterization has been known for general graphs. In this paper, we provide a characterization of the family of maximal barriers in general graphs, in which the known results are developed and unified.