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

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 et al. 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.