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 wellbehaved 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 is an element of N canonical tree-decompositions that distinguish all k-distinguishable ends efficiently. (c) 2021 Elsevier Inc. All rights reserved.
GraphCanonical tree of tree-decompositionTangleProfileEndGRAPHS
NSTL
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