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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.

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type ω + 1. Then we introduce and study a topological generalisation of infinite trees which can have limit edges, and show that every infinite tree set can be represented by the tree set admitted by a suitable such tree-like space.

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