We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combina- torial structures, which has further applications.
These include a new approach to cluster analysis and image segmentation. As another illustration for the abstract theorem, we show that applying it to edge-tangles yields the Gomory-Hu theorem.
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Diestel, R., Hundertmark, F. & Lemanczyk, S. Profiles of Separations: in Graphs, Matroids, and Beyond. Combinatorica 39, 37–75 (2019). https://doi.org/10.1007/s00493-017-3595-y
Mathematics Subject Classification (2010)