# Cyclewidth and the Grid Theorem for Perfect Matching Width of Bipartite Graphs

## TL;DR

This paper establishes a connection between perfect matching width and directed treewidth in bipartite graphs, confirming Norine's conjecture that high perfect matching width implies the presence of large grid minors.

## Contribution

It proves that for bipartite graphs, perfect matching width is equivalent to directed treewidth, linking it to the directed grid theorem and advancing the structural understanding of matching covered graphs.

## Key findings

- Perfect matching width equals directed treewidth in bipartite graphs.
- High perfect matching width implies the existence of large grid minors.
- Confirmed Norine's conjecture for bipartite graphs.

## Abstract

A connected graph G is called matching covered if every edge of G is contained in a perfect matching. Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition. It was introduced by Norine and intended as a tool for the structural study of matching covered graphs, especially in the context of Pfaffian orientations. Norine conjectured that graphs of high perfect matching width would contain a large grid as a matching minor, similar to the result on treewidth by Robertson and Seymour. In this paper we obtain the first results on perfect matching width since its introduction. For the restricted case of bipartite graphs, we show that perfect matching width is equivalent to directed treewidth and thus the Directed Grid Theorem by Kawarabayashi and Kreutzer for directed \treewidth implies Norine's conjecture.