Duality theorems for blocks and tangles in graphs

TL;DR

This paper establishes a broad duality theorem for tangles, blocks, and profiles in graphs, extending classical results and solving longstanding open problems in graph theory.

Contribution

It generalizes the classical tangle duality theorem to a wider range of graph structures, including profiles and blocks, addressing open questions in the field.

Findings

Proves a duality theorem for tangles, blocks, and profiles in graphs.

Extends classical tangle duality to new graph invariants.

Solves open problems posed by Diestel, Oum, and others.

Abstract

We prove a duality theorem applicable to a a wide range of specialisations, as well as to some generalisations, of tangles in graphs. It generalises the classical tangle duality theorem of Robertson and Seymour, which says that every graph either has a large-order tangle or a certain low-width tree-decomposition witnessing that it cannot have such a tangle. Our result also yields duality theorems for profiles and for $k$-blocks. This solves a problem studied, but not solved, by Diestel and Oum and answers an earlier question of Carmesin, Diestel, Hamann and Hundertmark.