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.
Mathematics Subject Classification (2010)
05C05 05C40 05C83 05B35 06A07
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J. Carmesin, R. Diestel, M. Hamann and F. Hundertmark: Canonical tree-decompositions of finite graphs I. Existence and algorithms, J. Combin. Theory Ser. B116 (2016), 1–24.MathSciNetCrossRefzbMATHGoogle Scholar
J. Carmesin, R. Diestel, M. Hamann and F. Hundertmark: Canonical tree-decompositions of finite graphs II. Essential parts, J. Combin. Theory Ser. B118 (2016), 268–283.MathSciNetCrossRefzbMATHGoogle Scholar