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Refining trees of tangles in abstract separation systems: inessential parts

Sandra Albrechtsen

TL;DR

This paper generalizes Erde's unifying tree-decomposition theorem for tangles from graphs to abstract, submodular separation systems. It develops a framework of canonical and refined tree-like witnesses (S-trees) for families of tangles, and proves refinements that render all inessential parts too small to host a tangle while preserving canonicity under automorphisms. A key contribution is the refinement lemma for inessential stars in arbitrary submodular separation systems and the construction of nested sets of good separations that distinguish all tangles in a canonical (though slightly weaker) sense. The results bridge tangle theory with abstract combinatorial structures, enabling structured insights into the organization of highly cohesive substructures beyond graphs. The work also outlines avenues for further refinement of both essential and inessential parts and their implications for canonical decompositions in broader contexts.

Abstract

Robertson and Seymour proved two fundamental theorems about tangles in graphs: the tree-of-tangles theorem, which says that every graph has a tree-decomposition such that distinguishable tangles live in different nodes of the tree, and the tangle-tree duality theorem, which says that graphs without a $k$-tangle have a tree-decomposition that witnesses the non-existence of such tangles, in that $k$-tangles would have to live in a node but no node is large enough to accommodate one. Erde combined these two fundamental theorems into one, by constructing a single tree-decomposition such that every node either accommodates a single $k$-tangle or is too small to accommodate one. Such a tree-decomposition thus shows at a glance how many $k$-tangles a graph has and where they are. The two fundamental theorems have since been extended to abstract separation systems, which support tangles in more general discrete structures. In this paper we extend Erde's unified theorem to such general systems.

Refining trees of tangles in abstract separation systems: inessential parts

TL;DR

This paper generalizes Erde's unifying tree-decomposition theorem for tangles from graphs to abstract, submodular separation systems. It develops a framework of canonical and refined tree-like witnesses (S-trees) for families of tangles, and proves refinements that render all inessential parts too small to host a tangle while preserving canonicity under automorphisms. A key contribution is the refinement lemma for inessential stars in arbitrary submodular separation systems and the construction of nested sets of good separations that distinguish all tangles in a canonical (though slightly weaker) sense. The results bridge tangle theory with abstract combinatorial structures, enabling structured insights into the organization of highly cohesive substructures beyond graphs. The work also outlines avenues for further refinement of both essential and inessential parts and their implications for canonical decompositions in broader contexts.

Abstract

Robertson and Seymour proved two fundamental theorems about tangles in graphs: the tree-of-tangles theorem, which says that every graph has a tree-decomposition such that distinguishable tangles live in different nodes of the tree, and the tangle-tree duality theorem, which says that graphs without a -tangle have a tree-decomposition that witnesses the non-existence of such tangles, in that -tangles would have to live in a node but no node is large enough to accommodate one. Erde combined these two fundamental theorems into one, by constructing a single tree-decomposition such that every node either accommodates a single -tangle or is too small to accommodate one. Such a tree-decomposition thus shows at a glance how many -tangles a graph has and where they are. The two fundamental theorems have since been extended to abstract separation systems, which support tangles in more general discrete structures. In this paper we extend Erde's unified theorem to such general systems.
Paper Structure (10 sections, 32 theorems, 8 equations, 5 figures)

This paper contains 10 sections, 32 theorems, 8 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Abstract separation systems and tangles
  3. Separation systems
  4. Profiles and tangles in abstract separation systems
  5. Tree sets and S-trees
  6. Tangle-tree duality
  7. Refining inessential stars
  8. Refining a canonical tree of tangles
  9. A tree of tangles with good separations
  10. Outlook

Key Result

Theorem 1

Let $\vS$ be a submodular separation system, and let $\mathcal{F}$ be a friendly canonical set of stars in $\vS$. Then there are nested sets $\tilde{N} \subseteq N \subseteq S$ such that:

Figures (5)

  • Figure 1: A tree-decomposition of $G$ that distinguishes all its $k$-tangles, and a refinement of the inessential part.
  • Figure 2: The $\vS_k(G)$-tree over $\bar{\mathcal{F}}$; the labeled leafs correspond to separations that are forced by $\bar{\mathcal{F}}$ but not by $\mathcal{F}$.
  • Figure 3: A visualization of \ref{['constr:Tc']}.
  • Figure 4: Sketch for the proof of \ref{['lem:NestedMaxSepsAreClRel']}.
  • Figure 5: A separation system and two profiles without a canonical good tree of tangles.

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Lemma 3.1
  • Proposition 3.2
  • ...and 43 more