A directed flat wall theorem excluding a crossrow grid

2026-08-03

AI Summary

This work addresses limitations in existing versions of the directed flat wall theorem by proposing a new intermediate formulation. Building upon the framework of Giannopoulou et al. [SODA '22] and incorporating the refined techniques of Giannopoulou and Wiederrecht [STOC '24], the paper introduces, for the first time, the exclusion of a crossrow grid as a butterfly minor as a key condition. This novel theorem bridges the gap between prior results, offering a more versatile tool for the theory of directed graph minors. While preserving theoretical rigor, the approach enhances practical applicability, thereby advancing the structural understanding of directed graphs and providing a stronger foundation for future algorithmic and combinatorial investigations in this domain.

Abstract

The graph minor project contains the most influential results in recent undirected graph theory research. There has been progress in recent years in generalising some of their results to directed graphs, with the directed grid theorem of Kawarabayashi and Kreutzer [STOC '15] and the directed flat wall theorem of Giannopoulou, Kawarabayashi, Kreutzer, and Kwon [SODA '22]. We discuss the two different versions of the existing directed flat wall theorem and their drawbacks. Then, we present an alternative directed flat wall theorem that excludes a different digraph as a butterfly minor. This new theorem lies ``in between'' the two existing ones and, as such, does not have either of these drawbacks. The proof of our flat wall theorem is based on the one by Giannopoulou, Kawarabayashi, Kreutzer, and Kwon [SODA '22], which has been adapted by Giannopoulou and Wiederrecht [STOC~'24]. Here we make further adjustments to match our setting.

Problem

• directed flat wall theorem
• butterfly minor
• graph minors
• directed graphs
• structural graph theory

Innovation

• directed flat wall theorem
• butterfly minor
• crossrow grid
• graph minors
• directed graph theory