Tangle structure trees II: trees of tangles and tangle-tree duality

TL;DR

This paper extends tangle structure trees to include new types of tangles and demonstrates how they can be used to derive and enhance fundamental theorems relating to graph and data set decompositions.

Contribution

It introduces generalized tangle structure trees for broader classes of tangles and connects these to classical tree-decomposition theorems, expanding their applicability.

Findings

Extended the tree-of-tangles theorem to $\\mathcal{F}$-tangles.

Converted tangle structure trees into classical tree-decompositions.

Provided certificates for the non-existence of certain tangles.

Abstract

Tangle structure trees, introduced in [3], offer a unified data structure that displays all the tangles of a graph or data set together with certificates for the non-existence of any other tangles, either locally or overall. In this paper we apply tangle structure trees to derive new versions of the two fundamental tangle theorems: the tree-of-tangles theorem, and the tangle-tree duality theorem. We extend the tree-of-tangles theorem to $\mathcal F$-tangles that need not be profiles. When $\mathcal F$ consists of stars of separations, as it does in classical tangle-tree duality theorems, we show how to convert tangle structure trees that certify the non-existence of $\mathcal F$-tangles into tree-decompositions that certify this in the way known from graph tangles, as $S$-trees over~$\mathcal F$.