Document Open Access Logo

Tangles and Single Linkage Hierarchical Clustering

Author Eva Fluck

Thumbnail PDF


  • Filesize: 457 kB
  • 12 pages

Document Identifiers

Author Details

Eva Fluck
  • RWTH Aachen University, Germany

Cite AsGet BibTex

Eva Fluck. Tangles and Single Linkage Hierarchical Clustering. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 38:1-38:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We establish a connection between tangles, a concept from structural graph theory that plays a central role in Robertson and Seymour’s graph minor project, and hierarchical clustering. Tangles cannot only be defined for graphs, but in fact for arbitrary connectivity functions, which are functions defined on the subsets of some finite universe, which in typical clustering applications consists of points in some metric space. Connectivity functions are usually required to be submodular. It is our first contribution to show that the central duality theorem connecting tangles with hierarchical decompositions (so-called branch decompositions) also holds if submodularity is replaced by a different property that we call maximum-submodular. We then define a natural, though somewhat unusual connectivity function on finite data sets in an arbitrary metric space and prove that its tangles are in one-to-one correspondence with the clusters obtained by applying the well-known single linkage clustering algorithms to the same data set. The idea of viewing tangles as clusters has first been proposed by Diestel and Whittle [Reinhard Diestel et al., 2019] as an approach to image segmentation. To the best of our knowledge, our result is the first that establishes a precise technical connection between tangles and clusters.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Unsupervised learning and clustering
  • Tangles
  • Branch Decomposition
  • Duality
  • Hierarchical Clustering
  • Single Linkage


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Isolde Adler, Georg Gottlob, and Martin Grohe. Hypertree width and related hypergraph invariants. European Journal of Combinatorics, 28(8):2167-2181, 2007. URL:
  2. Gunnar Carlsson and Facundo Mémoli. Characterization, stability and convergence of hierarchical clustering methods. Journal of Machine Learning Research, 11(Apr):1425-1470, 2010. Google Scholar
  3. Vincent Cohen-Addad, Varun Kanade, Frederik Mallmann-Trenn, and Claire Mathieu. Hierarchical Clustering: Objective Functions and Algorithms. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 378-397. SIAM, 2018. URL:
  4. Sanjoy Dasgupta. A cost function for similarity-based hierarchical clustering. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 118-127. ACM, 2016. URL:
  5. Reinhard Diestel, Fabian Hundertmark, and Sahar Lemanczyk. Profiles of Separations: in Graphs, Matroids, and Beyond. Combinatorica, 39(1):37-75, 2019. URL:
  6. Reinhard Diestel and Sang-il Oum. Unifying Duality Theorems for Width Parameters in Graphs and Matroids (Extended Abstract). In Dieter Kratsch and Ioan Todinca, editors, Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Nouan-le-Fuzelier, France, June 25-27, 2014. Revised Selected Papers, volume 8747 of Lecture Notes in Computer Science, pages 1-14. Springer, 2014. URL:
  7. Reinhard Diestel and Geoff Whittle. Tangles and the Mona Lisa. ArXiv, 2016. arXiv:1603.06652 [math.CO]. URL:
  8. Jim Geelen, Bert Gerards, and Geoff Whittle. Tangles, tree-decompositions and grids in matroids. Journal of Combinatorial Theory, Series B, 99(4):657-667, 2009. URL:
  9. Martin Grohe. Tangled up in blue (a survey on connectivity, decompositions, and tangles). ArXiv, 2016. arXiv:1605.06704 [cs.DM]. Google Scholar
  10. Jon M Kleinberg. An impossibility theorem for clustering. In Suzanna Becker, Sebastian Thrun, and Klaus Obermayer, editors, Advances in Neural Information Processing Systems 15 [Neural Information Processing Systems, NIPS 2002, December 9-14, 2002, Vancouver, British Columbia, Canada], pages 446-453. MIT Press, 2002. URL:
  11. Neil Robertson and Paul D Seymour. Graph minors. X. Obstructions to tree-decomposition. Journal of Combinatorial Theory, Series B, 52(2):153-190, 1991. URL:
  12. Ulrike von Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4):395-416, 2007. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail