Excluding A Grid Minor In Planar Digraphs

TL;DR

This paper proves that planar digraphs with large directed tree-width necessarily contain a large cylindrical grid minor, confirming a conjecture linking tree-width and grid minors in directed graphs.

Contribution

It establishes the conjecture that large directed tree-width implies the existence of a large cylindrical grid minor in planar digraphs, extending the understanding of graph minors.

Findings

Proved the conjecture for planar digraphs.

Many proof steps are applicable to general digraphs.

Provides foundational results for directed graph minor theory.

Abstract

In [Directed tree-width, J. Combin. Theory Ser. B 82 (2001), 138-154] we introduced the notion of tree-width of directed graphs and presented a conjecture, formulated during discussions with Noga Alon and Bruce Reed, stating that a digraph of huge tree-width has a large "cylindrical grid" minor. Here we prove the conjecture for planar digraphs, but many steps of the proof work in general. This is an unedited and unpolished manuscript from October 2001. Since many people asked for copies we are making it available in the hope that it may be useful. The conjecture was proved by Kawarabayashi and Kreutzer in arXiv:1411.5681.