Tangle-Tree Duality: In Graphs, Matroids and Beyond

Abstract

We apply a recent tangle-tree duality theorem in abstract separation systems to derive tangle-tree-type duality theorems for width-parameters in graphs and matroids.We further derive a duality theorem for the existence of clusters in large data sets.

Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a tangle-type duality theorem for tree-width.

Our results can also be used to derive short proofs of all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width and rank-width.

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Correspondence to Reinhard Diestel or Sang-il Oum.

Additional information

Supported by IBS-R029-C1 and the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. NRF-2017R1A2B4005020).

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Diestel, R., Oum, Si. Tangle-Tree Duality: In Graphs, Matroids and Beyond. Combinatorica 39, 879–910 (2019). https://doi.org/10.1007/s00493-019-3798-5

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Mathematics Subject Classification (2010)

  • 05C40
  • 05C75
  • 05C83