A Flat Wall Theorem for Matching Minors in Bipartite Graphs

TL;DR

This paper establishes a matching minor analogue of the Flat Wall Theorem for bipartite graphs excluding a fixed matching minor, linking structural graph theory and matching theory to extend results to digraphs.

Contribution

It introduces a novel Flat Wall Theorem for bipartite graphs with matching minors, bridging structural and matching theories, and extends to digraphs.

Findings

Proves a matching minor version of the Flat Wall Theorem for bipartite graphs.

Establishes a relationship between structural digraph theory and matching theory.

Provides a new Flat Wall Theorem for digraphs that differs from previous variants.

Abstract

A major step in the graph minors theory of Robertson and Seymour is the transition from the Grid Theorem which, in some sense uniquely, describes areas of large treewidth within a graph, to a notion of local flatness of these areas in form of the existence of a large flat wall within any huge grid of an H-minor free graph. In this paper, we prove a matching theoretic analogue of the Flat Wall Theorem for bipartite graphs excluding a fixed matching minor. Our result builds on a a tight relationship between structural digraph theory and matching theory and allows us to deduce a Flat Wall Theorem for digraphs which substantially differs from a previously established directed variant of this theorem.