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Tangle structure trees

Hanno von Bergen, Reinhard Diestel

TL;DR

This work introduces tangle structure trees as a unified, constructive data structure that simultaneously represents all $\mathcal{F}$-tangles of an abstract separation system and provides actionable certificates for non-existence or non-extendability of tangles. It generalizes the tree-of-tangles and tangle-tree duality to arbitrary obstruction sets $\mathcal{F}$ by introducing standard and rich conditions, and proves the existence of efficient, irreducible, ordered $\mathcal{F}$-tangle structure trees. The framework yields two complementary outcomes: (i) identifying all tangles via a single tree with leaves encoding tangles and internal nodes ensuring no $\mathcal{F}$-subset, and (ii) producing $\mathcal{F}$-trees that certify non-existence when tangles are absent, with applicability to $S_k$-restricted tangles. The results extend to blocks, profiles, and cluster tangles, including non-star obstruction sets like $\mathcal{B}_k$, $\mathcal{P}_s$, and $\mathcal{C}_n$, and are accompanied by constructive proofs and open-source software for tangle detection and analysis, enabling practical use in graphs and large datasets.

Abstract

We introduce a comprehensive data structure, tangle structure trees, which simultaneously displays all the $\mathcal{F}$-tangles of an abstract separation system for very general obstruction sets $\mathcal{F}$. It simultaneously also displays certificates $σ\in\mathcal{F}$ for any non-existence of such tangles, or for the non-extendability of low-order tangles to higher-order ones. Our theorem can be applied to produce the structures of the classical tree-of-tangles and tangle-tree duality theorems, both for graph tangles and for their known generalizations to more general separation systems. It extends those theorems to obstruction sets $\mathcal{F}$ that need not define profiles (as they must in trees of tangles) or consist of stars of separations (as they must in tangle-tree duality). Our existence proof for these structure trees is constructive. The construction has been implemented in open-source software available for tangle detection and further analysis.

Tangle structure trees

TL;DR

This work introduces tangle structure trees as a unified, constructive data structure that simultaneously represents all -tangles of an abstract separation system and provides actionable certificates for non-existence or non-extendability of tangles. It generalizes the tree-of-tangles and tangle-tree duality to arbitrary obstruction sets by introducing standard and rich conditions, and proves the existence of efficient, irreducible, ordered -tangle structure trees. The framework yields two complementary outcomes: (i) identifying all tangles via a single tree with leaves encoding tangles and internal nodes ensuring no -subset, and (ii) producing -trees that certify non-existence when tangles are absent, with applicability to -restricted tangles. The results extend to blocks, profiles, and cluster tangles, including non-star obstruction sets like , , and , and are accompanied by constructive proofs and open-source software for tangle detection and analysis, enabling practical use in graphs and large datasets.

Abstract

We introduce a comprehensive data structure, tangle structure trees, which simultaneously displays all the -tangles of an abstract separation system for very general obstruction sets . It simultaneously also displays certificates for any non-existence of such tangles, or for the non-extendability of low-order tangles to higher-order ones. Our theorem can be applied to produce the structures of the classical tree-of-tangles and tangle-tree duality theorems, both for graph tangles and for their known generalizations to more general separation systems. It extends those theorems to obstruction sets that need not define profiles (as they must in trees of tangles) or consist of stars of separations (as they must in tangle-tree duality). Our existence proof for these structure trees is constructive. The construction has been implemented in open-source software available for tangle detection and further analysis.
Paper Structure (7 sections, 28 theorems, 13 equations, 1 figure)

This paper contains 7 sections, 28 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tangles basics
  3. Tangle structure trees
  4. Existence of tangle structure trees
  5. Irreducible and efficient tangle structure trees
  6. The $\mathcal{F}$-tangle structure theorems
  7. Applications: blocks, profiles, and cluster tangles

Key Result

Lemma 2.1

Let $\sigma\subseteq\tau\subseteq{ {{{\mathop{ S}\limits^{\hbox{$\rightarrow$}}}}} }$ be consistent sets. Then $\lfloor\sigma\rfloor\subseteq\lfloor\tau\rfloor$. If $\tau$ is an orientation of all of $S$, then $\lfloor \tau \rfloor = \tau$.

Figures (1)

  • Figure 1: Nested separations $r = \{A,B\}$ and $s = \{C,D\}$ of a graph. Their orientations ${\mathop{ r}\limits^{\hbox{$\rightarrow$}}} = (A,B)$ and ${{ {\mathop{ s}\limits^{\hbox{$\leftarrow$}}}} } = (D,C)$ point towards each other, since ${\mathop{ r}\limits^{\hbox{$\rightarrow$}}}\ge {\mathop{ s}\limits^{\hbox{$\rightarrow$}}}$ (as $B\supseteq D$) and ${{ {\mathop{ s}\limits^{\hbox{$\leftarrow$}}}} }\ge{\mathop{ r}\limits^{\hbox{$\leftarrow$}}}$ (as $C\supseteq A$).

Theorems & Definitions (57)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 47 more