Potatoes and Onions

Marcin Briański; Meike Hatzel; Karolina Okrasa; Michał Pilipczuk

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arXiv · 2610.20391

Potatoes and Onions

Marcin Briański·Meike Hatzel·Karolina Okrasa·Michał Pilipczuk

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Abstract

The first major step in the theory of graph minors of Robertson and Seymour was the grid theorem, a fundamental duality result describing that any graph of sufficiently high treewidth must contain a large grid as a minor. Since this pioneering work, multiple grid theorems have been established in structural graph theory, linking various width parameters of combinatorial objects with the non-existence of grid-like obstructions. In particular, Kawarabayashi and Kreutzer [STOC'15] proved a variant of the grid theorem for directed graphs that links directed treewidth with the non-existence of topological minors of large cylindrical walls. On the other hand, to study immersions of undirected graphs $-$ another containment relation that focuses on edge-disjointness, rather than vertex-disjointness $-$ Wollan [JCTB'15] introduced a parameter, tree-cutwidth, and proved that it is dual to immersions of large walls. In this work, we present a version of Wollan's parameter for directed graphs, the directed tree-cutwidth. The main result is a grid theorem for directed tree-cutwidth, establishing that it is dual to directed immersions of large cylindrical walls.

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Marcin Briański, Meike Hatzel, Karolina Okrasa, Michał Pilipczuk. 2026-10-14. Potatoes and Onions. https://arxiv.org/abs/2610.20391

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