Cycles of Well-Linked Sets II: an Elementary Bound for the Directed Grid Theorem

TL;DR

This paper provides a simpler, more modular proof of the Directed Grid Theorem, significantly improving the upper bound for the function relating directed treewidth to the existence of large cylindrical grid minors.

Contribution

An alternative, simpler proof of the Directed Grid Theorem with a reduced upper bound for the bounding function, introducing the concept of cycles of well-linked sets (CWS).

Findings

The new proof is conceptually simpler and more modular.

The upper bound for the function f is improved to a power tower of height 22.

Large directed treewidth guarantees the existence of large cylindrical grid minors.

Abstract

In 2015, Kawarabayashi and Kreutzer proved the Directed Grid Theorem - the generalisation of the well-known Excluded Grid Theorem to directed graphs - confirming a conjecture by Reed, Johnson, Robertson, Seymour and Thomas from the mid-nineties. The theorem states that there is a function $f$ such that every digraph of directed treewidth $f(k)$ contains a cylindrical grid of order $k$ as a butterfly minor. However, the given function grows faster than any non-elementary function of the size of the grid minor. More precisely, it is larger than a power tower whose height depends on the size of the grid. In this paper, we present an alternative proof of the Directed Grid Theorem which is conceptually much simpler, more modular in composition and improves the upper bound for the function $f$ to a power tower of height $22$. A key concept of our proof is a new structure called cycles of well-linked sets (CWS). We show that any digraph of large directed treewidth contains a large CWS, which in turn contains a large cylindrical grid.