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arXiv · 2304.12078

Optimal trees of tangles: refining the essential parts

Sandra Albrechtsen
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Abstract

We combine the two fundamental fixed-order tangle theorems of Robertson and Seymour into a single theorem that implies both, in a best possible way. We show that, for every $k \in \mathbb{N}$, every tree-decomposition of a graph $G$ which efficiently distinguishes all its $k$-tangles can be refined to a tree-decomposition whose parts are either too small to be home to a $k$-tangle, or as small as possible while being home to a $k$-tangle.

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Sandra Albrechtsen. 2026-07-01. Optimal trees of tangles: refining the essential parts. https://arxiv.org/abs/2304.12078

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