Canonical trees of tree-decompositions
2020
We prove that every graph has a canonical tree of tree-decompositions that distinguishes all 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 Kron as well as of Carmesin, Diestel, Hundertmark and Stein. Beyond that for locally finite graphs our result gives for each $k\in\mathbb N$ canonical tree-decompositions that distinguish all $k$-distinguishable ends efficiently.
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