Tangle structure trees II: trees of tangles and tangle-tree duality
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917351651278848 |
|---|---|
| author | von Bergen, Hanno Diestel, Reinhard |
| author_facet | von Bergen, Hanno Diestel, Reinhard |
| contents | 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~$\mathcal F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17756 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tangle structure trees II: trees of tangles and tangle-tree duality von Bergen, Hanno Diestel, Reinhard Combinatorics 05C83 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~$\mathcal F$. |
| title | Tangle structure trees II: trees of tangles and tangle-tree duality |
| topic | Combinatorics 05C83 |
| url | https://arxiv.org/abs/2603.17756 |