Refining a Tree-Decomposition which Distinguishes Tangles

TL;DR

This paper improves the understanding of canonical tree-decompositions by showing that inessential parts have torsos with branch width less than k, refining the decomposition structure for highly connected graph parts.

Contribution

It demonstrates that inessential parts in canonical tree-decompositions have torsos with branch width less than k, enabling further refinement of the decomposition.

Findings

Inessential parts have torsos with branch width less than k.

Refinement of canonical tree-decompositions is possible based on this property.

Similar results hold for k-blocks.

Abstract

Roberston and Seymour introduced tangles of order $k$ as objects representing highly connected parts of a graph and showed that every graph admits a tree-decomposition of adhesion $<k$ in which each tangle of order $k$ is contained in a different part. Recently, Carmesin, Diestel, Hamann and Hundertmark showed that such a tree-decomposition can be constructed in a canonical way, which makes it invariant under automorphisms of the graph. These canonical tree-decompositions necessarily have parts which contain no tangle of order $k$. We call these parts inessential. Diestel asked what could be said about the structure of the inessential parts. In this paper we show that the torsos of the inessential parts in these tree-decompositions have branch width $<k$, allowing us to further refine the canonical tree-decompositions, and also show that a similar result holds for $k$-blocks.