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We construct tree-decompositions of graphs that distinguish all theirk-blocks and tangles of order k, for any fixed integer k. We describe a familyof algorithms to construct such decompositions, seeking to maximize theirdiversity subject to the requirement that they commute with graph isomorphisms.In particular, all the decompositions constructed are invariant under theautomorphisms of the graph.

Canonical tree-decompositions of finite graphs I. Existence and algorithms