Canonical trees of tree-decompositions

We prove that every graph has a canonical tree of tree-decompositions that distinguishes all principal tangles (these include the ends and various kinds of large finite dense structures) efficiently.

Here ‘trees of tree-decompositions’ are a slightly weaker notion than ‘tree-decompositions’ but much more well-behaved than ‘tree-like metric spaces’. This theorem is best possible in the sense that we give an example that ‘trees of tree-decompositions’ cannot be strengthened to ‘tree-decompositions’ in the above theorem.

This implies results of Dunwoody and Krön as well as of Carmesin, Diestel, Hundertmark and Stein. Beyond that for locally finite graphs our result gives for each k ε ℕ canonical tree-decompositions that distinguish all k-distinguishable ends efficiently.