Optimal trees of tangles: refining the essential parts

TL;DR

This paper unifies two key tangle theorems into a single, optimal theorem, enabling refined tree-decompositions that precisely distinguish and contain graph tangles efficiently.

Contribution

It introduces a unified theorem that refines existing tangle theorems, optimizing tree-decompositions for better graph tangle analysis.

Findings

Unified theorem implies both fundamental tangle theorems.

Refined tree-decompositions distinguish all $k$-tangles efficiently.

Parts of the refined decomposition are optimally small or too small for tangles.

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.