Structure of Towers and a New Proof of the Tight Cut Lemma

Abstract

In the first part of our study, we extend the theory of basilica canonical decomposition by introducing new concepts known as towers and tower-sequences. The basilica canonical decomposition is a recently proposed tool in matching theory that can be applied non-trivially even for general graphs with perfect matchings. When studying matchings, the structure of alternating paths frequently needs to be considered. We show how a graph is made up of towers and tower-sequences, and thus obtain the structure of alternating paths in terms of the basilica canonical decomposition. This result provides a strong tool for analyzing general graphs with perfect matchings.

The second part of our study is a new graph theoretic proof of the so-called Tight Cut Lemma derived from the first part of our study. To derive a characterization of the perfect matchings polytope, Edmonds, Lovász, and Pulleyblank introduced the Tight Cut Lemma as the most challenging aspect of their work. The Tight Cut Lemma in fact claims bricks as the fundamental building blocks that constitute a graph and can be referred to as a key result in this field. Although the Tight Cut Lemma itself is a purely graph theoretic statement, there was no known graph theoretic proof for decades until Szigeti provided such a proof using Frank-Szigeti’s optimal ear decomposition theory.

By contrast, we provide a new proof using the extended theory of basilica canonical decomposition as the only preliminary result, and accordingly proposes a new strategy for studying bricks and tight cuts or matching theory in general. Our proof shows how the discussions on alternating paths construct the Tight Cut Lemma from first principles via the basilica canonical decomposition, even without using barriers, that is, the dual notion of matchings. The distinguishing features of our proof are that it is purely graph theoretic, purely matching (cardinality 1-matching) theoretic, and purely “primal” with respect to matchings.

Supported by JSPS KAKENHI Grant Number 15J09683.