A directed flat wall theorem excluding a crossrow grid
For researchers in graph theory, this provides a new directed flat wall theorem that fills a gap between existing results, potentially enabling further applications in directed graph minor theory.
The paper addresses drawbacks in existing directed flat wall theorems by presenting an alternative version that excludes a different digraph as a butterfly minor, lying between the two existing versions and avoiding their drawbacks. The proof adapts prior work by Giannopoulou et al. and Giannopoulou and Wiederrecht.
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.