Twins in Subdivision Drawings of Hypergraphs

  • René van Bevern
  • Iyad Kanj
  • Christian Komusiewicz
  • Rolf Niedermeier
  • Manuel Sorge
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)


Visualizing hypergraphs, systems of subsets of some universe, has continuously attracted research interest in the last decades. We study a natural kind of hypergraph visualization called subdivision drawings. Dinkla et al. [Comput. Graph. Forum ’12] claimed that only few hypergraphs have a subdivision drawing. However, this statement seems to be based on the assumption (also used in previous work) that the input hypergraph does not contain twins, pairs of vertices which are in precisely the same hyperedges (subsets of the universe). We show that such vertices may be necessary for a hypergraph to admit a subdivision drawing. As a counterpart, we show that the number of such “necessary twins” is upper-bounded by a function of the number m of hyperedges and a further parameter r of the desired drawing related to its number of layers. This leads to a linear-time algorithm for determining such subdivision drawings if m and r are constant; in other words, the problem is linear-time fixed-parameter tractable with respect to the parameters m and r.




  1. 1.
    Alsallakh, B., Micallef, L., Aigner, W., Hauser, H., Miksch, S., Rodgers, P.J.: The state-of-the-art of set visualization. Comput. Graph. Forum 35(1), 234–260 (2016)CrossRefGoogle Scholar
  2. 2.
    Beeri, C., Fagin, R., Maier, D., Yannakakis, M.: On the desirability of acyclic database schemes. J. ACM 30(3), 479–513 (1983)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bixby, R.E., Wagner, D.K.: An almost linear-time algorithm for graph realization. Math. Oper. Res. 13(1), 99–123 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brandes, U., Cornelsen, S., Pampel, B., Sallaberry, A.: Blocks of hypergraphs—applied to hypergraphs and outerplanarity. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 201–211. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-19222-7_21 CrossRefGoogle Scholar
  5. 5.
    Brandes, U., Cornelsen, S., Pampel, B., Sallaberry, A.: Path-based supports for hypergraphs. J. Discrete Algorithms 14, 248–261 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Buchin, K., van Kreveld, M.J., Meijer, H., Speckmann, B., Verbeek, K.: On planar supports for hypergraphs. J. Graph Algorithms Appl. 15(4), 533–549 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, J., Komusiewicz, C., Niedermeier, R., Sorge, M., Suchý, O., Weller, M.: Polynomial-time data reduction for the subset interconnection design problem. SIAM J. Discrete Math. 29(1), 1–25 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)CrossRefMATHGoogle Scholar
  9. 9.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Berlin (2010)CrossRefMATHGoogle Scholar
  10. 10.
    Dinkla, K., van Kreveld, M.J., Speckmann, B., Westenberg, M.A.: Kelp diagrams: point set membership visualization. Comput. Graph. Forum 31(3), 875–884 (2012)CrossRefGoogle Scholar
  11. 11.
    Eschbach, T., Günther, W., Becker, B.: Orthogonal hypergraph drawing for improved visibility. J. Graph Algorithms Appl. 10(2), 141–157 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Flower, J., Fish, A., Howse, J.: Euler diagram generation. J. Vis. Lang. Comput. 19(6), 675–694 (2008)CrossRefGoogle Scholar
  13. 13.
    Habib, M., Paul, C., Viennot, L.: Partition refinement techniques: an interesting algorithmic tool kit. Int. J. Found. Comput. Sci. 10(2), 147–170 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Johnson, D.S., Pollak, H.O.: Hypergraph planarity and the complexity of drawing Venn diagrams. J. Graph Theory 11(3), 309–325 (1987)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kaufmann, M., Kreveld, M., Speckmann, B.: Subdivision drawings of hypergraphs. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 396–407. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-00219-9_39 CrossRefGoogle Scholar
  16. 16.
    Klemz, B., Mchedlidze, T., Nöllenburg, M.: Minimum tree supports for hypergraphs and low-concurrency Euler diagrams. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 265–276. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-08404-6_23 CrossRefGoogle Scholar
  17. 17.
    Korach, E., Stern, M.: The clustering matroid and the optimal clustering tree. Math. Program. 98(1–3), 385–414 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mäkinen, E.: How to draw a hypergraph. Int. J. Comput. Math. 34, 178–185 (1990)CrossRefMATHGoogle Scholar
  19. 19.
    Tarjan, R., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Tutte, W.T.: How to draw a graph. Proc. London Math. Soc. 13(3), 743–768 (1963)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Verroust, A., Viaud, M.-L.: Ensuring the drawability of extended Euler diagrams for up to 8 sets. In: Blackwell, A.F., Marriott, K., Shimojima, A. (eds.) Diagrams 2004. LNCS (LNAI), vol. 2980, pp. 128–141. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-25931-2_13 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • René van Bevern
    • 1
    • 2
  • Iyad Kanj
    • 3
  • Christian Komusiewicz
    • 4
  • Rolf Niedermeier
    • 5
  • Manuel Sorge
    • 5
  1. 1.Novosibirsk State UniversityNovosibirskRussian Federation
  2. 2.Sobolev Institute of MathematicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussian Federation
  3. 3.DePaul UniversityChicagoUSA
  4. 4.Friedrich-Schiller-Universität JenaJenaGermany
  5. 5.TU BerlinBerlinGermany

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