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

Hanno von Bergen, Reinhard Diestel

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~$\mathcal F$.

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

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 -tangles that need not be profiles. When 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 -tangles into tree-decompositions that certify this in the way known from graph tangles, as -trees over~.
Paper Structure (7 sections, 29 theorems, 20 equations, 2 figures)

This paper contains 7 sections, 29 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Separation systems and their tangles
  3. Tangle structure trees
  4. Tangle-tree duality
  5. Submodular order functions
  6. Trees of tangles
  7. Zooming in: how to make consistent subsets more efficient

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 $\lfloor \tau \rfloor = \tau$. If $\tau$ has no elements that are co-trivial in ${ {{{\mat

Figures (2)

  • 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 $C\supseteq A$).
  • Figure 2: A corner ${\mathop{ t}\limits^{\hbox{$\rightarrow$}}} = {\mathop{ r}\limits^{\hbox{$\rightarrow$}}}\land {\mathop{ s}\limits^{\hbox{$\rightarrow$}}}$ of two bipartitions of a set $V\!$.

Theorems & Definitions (54)

  • Lemma 2.1: TSTs
  • Definition 2.2
  • Theorem 3.1: TSTs
  • Definition 3.2
  • Theorem 3.3: TSTs
  • Theorem 3.4: TSTs
  • Theorem 4.1: TSTs
  • Lemma 4.2: TreeSets
  • Theorem 4.3
  • Theorem 4.4
  • ...and 44 more