Trees of tangles in abstract separation systems

TL;DR

The paper develops a unifying, highly abstract framework for tree-of-tangles via separation systems, introducing the Splinter Lemma and its canonical variant to construct nested sets that efficiently distinguish profiles. It achieves short, purely structural proofs of classical results (e.g., GMX) and extends canonical tree-decomposition theory to broader separation types, including clique and circle separations. By connecting noncanonical and canonical perspectives through compatible sequences and hierarchical splintering, it broadens applicability to graphs, matroids, and other combinatorial settings while preserving canonicity where possible. The results substantially simplify proofs and broaden the scope of tree-of-tangles theorems in submodular and structurally submodular contexts.

Abstract

We prove canonical and non-canonical tree-of-tangles theorems for abstract separation systems that are merely structurally submodular. Our results imply all known tree-of-tangles theorems for graphs, matroids and abstract separation systems with submodular order functions, with greatly simplified and shortened proofs.

Figures (2)

• Figure 1: Robustness.
• Figure 2: The possible configurations in \ref{['property:hierarchically2_equal']} in the definition of splinter hierarchically, up to symmetry.

Theorems & Definitions (43)

• Theorem 1.1: GMX
• Theorem 1.2: confing
• Theorem 1.3: AbstractTangles*Theorem 6
• Theorem 1.4
• Lemma 1.4: Splinter Lemma
• Lemma 1.4: Canonical Splinter Lemma
• Lemma 3.1: ProfilesNew*Lemma 2.1
• Lemma 3.1: Splinter Lemma
• proof
• Theorem 4.1: GMX