Canonical tree-decompositions of a graph that display its $k$-blocks

TL;DR

This paper constructs canonical tree-decompositions for finite graphs that precisely display all separable $k$-blocks, confirming a conjecture and enhancing understanding of graph structure related to $k$-blocks and tangles.

Contribution

It provides a method to build canonical tree-decompositions where each separable $k$-block is uniquely represented, confirming a conjecture of Diestel.

Findings

Constructed canonical tree-decompositions for all $k$-blocks.

Each separable $k$-block is uniquely represented as a part.

Confirmed a conjecture of Diestel regarding $k$-block display.

Abstract

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by removing less than $k$ vertices. It is separable if there exists a tree-decomposition of adhesion less than $k$ of $G$ in which this $k$-block appears as a part. Carmesin, Diestel, Hamann, Hundertmark and Stein proved that every finite graph has a canonical tree-decomposition of adhesion less than $k$ that distinguishes all its $k$-blocks and tangles of order $k$. We construct such tree-decompositions with the additional property that every separable $k$-block is equal to the unique part in which it is contained. This proves a conjecture of Diestel.