Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Obtaining trees of tangles from tangle-tree duality

Christian Elbracht, Jay Lilian Kneip, Maximilian Teegen

TL;DR

The paper unifies tree-of-tangles results with the tangle-tree duality theorem in abstract separation systems, enabling derivations of tree structures that distinguish profile collections and providing degree-bounding capabilities. It strengthens the tangle-tree duality framework and develops tools such as splices, shifting, and efficient distinguishers to construct nested separator trees that efficiently separate multiple profiles, including those of mixed orders. The authors introduce a sequence of refinements—from structurally submodular to distributive universes and strongly robust profiles—to extend tree-of-tangles results to tangles of different orders and to bound node degrees. The work thereby bridges core pillars of tangle theory, offering practical methods to obtain trees of tangles with controlled complexity across orders and settings, with implications for generalizations beyond graphs and matroids.

Abstract

We demonstrate the versatility of the tangle-tree duality theorem for abstract separation systems by using it to prove tree-of-tangles theorems. This approach allows us to strengthen some of the existing tree-of-tangles theorems by bounding the node degrees in them. We also present a slight strengthening and simplified proof of the duality theorem, which allows us to derive a tree-of-tangles theorem also for tangles of different orders.

Obtaining trees of tangles from tangle-tree duality

TL;DR

The paper unifies tree-of-tangles results with the tangle-tree duality theorem in abstract separation systems, enabling derivations of tree structures that distinguish profile collections and providing degree-bounding capabilities. It strengthens the tangle-tree duality framework and develops tools such as splices, shifting, and efficient distinguishers to construct nested separator trees that efficiently separate multiple profiles, including those of mixed orders. The authors introduce a sequence of refinements—from structurally submodular to distributive universes and strongly robust profiles—to extend tree-of-tangles results to tangles of different orders and to bound node degrees. The work thereby bridges core pillars of tangle theory, offering practical methods to obtain trees of tangles with controlled complexity across orders and settings, with implications for generalizations beyond graphs and matroids.

Abstract

We demonstrate the versatility of the tangle-tree duality theorem for abstract separation systems by using it to prove tree-of-tangles theorems. This approach allows us to strengthen some of the existing tree-of-tangles theorems by bounding the node degrees in them. We also present a slight strengthening and simplified proof of the duality theorem, which allows us to derive a tree-of-tangles theorem also for tangles of different orders.

Paper Structure

This paper contains 13 sections, 35 theorems, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Terminology and background
  3. Basic definitions of abstract separation systems
  4. The tree-of-tangles theorem
  5. Tangle-tree duality
  6. Splices in submodular universes
  7. Structurally submodular separation systems
  8. Application: Degrees in trees of tangles
  9. Efficient distinguishers
  10. Degrees in efficient trees of tangles
  11. Tangles of mixed orders
  12. A Short Adventure into Duality
  13. Obtaining a tree-of-tangles theorem for different order tangles from tangle-tree duality

Key Result

Theorem 1.1

Let $\vS$ be a structurally submodular separation system and $\mathcal{P}$ a set of profiles of $S$. Then $\vS$ contains a tree set that distinguishes $\mathcal{P}$.

Figures (3)

  • Figure 1: A ground-set and system of bipartitions.
  • Figure :
  • Figure :

Theorems & Definitions (55)

  • Theorem 1.1: name=AbstractTangles*Theorem 6, restate=totDaniel
  • Theorem 1.2: Tangle-tree duality theorem TangleTreeAbstract*Theorem 4.3
  • Lemma 2.1: AbstractSepSys*Lemma 3.2
  • Theorem 2.2: ProfilesNew*Corollary 3.7, modified
  • Theorem 2.2: Tangle-tree duality theorem TangleTreeAbstract*Theorem 4.3
  • Lemma 2.3: TangleTreeAbstract*Lemma 2.1
  • Lemma 2.4: TangleTreeAbstract*Lemma 2.2
  • Lemma 2.5: TangleTreeAbstract*Lemma 2.3
  • Lemma 2.6: TangleTreeAbstract*Lemma 2.4
  • Lemma 2.7: AbstractTangles*Lemma 13
  • ...and 45 more