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

TL;DR

The Directed Grid Theorem for digraphs is advanced by providing an elementary, modular proof that improves the known bounds: there exists a bound $f(k)$ such that any digraph with directed treewidth at least $f(k)$ contains a cylindrical grid of order $k$ as a butterfly minor, with $f(k)$ bounded by a power tower of height $22$ (i.e., $f(k) \in\, ext{exp}_2^{22}( ext{poly}^9(k))$). The authors introduce cycles of well-linked sets (CWS) and a 2-horizontal web framework to transform large directed treewidth into structured connectivity that yields cylindrical grids via a sequence of intermediate objects (paths of well-linked/ordered sets, back-linkages, fences, and webs). The key contributions include an elementary bound on the grid minor function, a modular proof architecture that isolates the back-linkage and web-building stages, and an explicit connection to Erdős–Pósa-type results in directed graphs through tighter dtw-to-grid implications. The work advances the practical applicability of the Directed Grid Theorem by reducing the non-elementary blow-up and clarifying the structural steps needed to approach even tighter bounds, with significant implications for directed graph structure theory and related algorithmic results.

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.

Figures (11)

• Figure 1: A cylindrical grid of order 3 on the left and the cylindrical wall of order 3 on the right. The split vertices on the wall have a grey background, where the cyan pentagon is the vertex $v^{\text{out}}$ and the blue rectangle is the vertex $v^{\text{in}}$.
• Figure 2: How a subdivision of the cylindrical wall of order 3 can be constructed inside a cylindrical grid of order 4. Unused arcs and vertices of the grid are coloured grey. The split vertices of the wall have a grey background, where the blue square vertex represents $v^{\text{in}}$ and the cyan pentagon vertex represents $v^{\text{out}}$.
• Figure 3: A fence and, on the right, an illustration of how a cylindrical grid is obtained from a fence. The dotted orange paths symbolise the arcs $e_i$ that close the cycles drawn solid in \ref{['fig:cylindrical-grid']}.
• Figure 4: An acyclic grid and a fence.
• Figure 5: The layers $D_j(T)$ of the temporal graph $T \coloneqq \left( V = \Set{a,b,c}, \mathcal{A} = \Set{A_1,A_2,A_3} \right)$ constructed from the graphs $Q_j$ displayed above as defined in \ref{['def:routing-temporal-digraph']}.

Theorems & Definitions (89)

• Theorem 1.0: COSSI
• Theorem 1.0
• Theorem 1.0
• Theorem 1.1
• Lemma 1.2: amiri2016erdos
• Theorem 1.3
• proof
• Theorem 2.1: erdosszekeres1935
• Theorem 2.2: Menger's Theorem menger
• Lemma 2.3: kawarabayashi2015directed