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Simultaneous Embeddability of Two Partitions

  • Jan Christoph Athenstädt
  • Tanja Hartmann
  • Martin Nöllenburg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)

Abstract

We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous embeddability (weak, strong, and full embeddability) that differ by increasingly strict well-formedness conditions on how different block regions are allowed to intersect. We show that these simultaneous embeddability classes are closely related to different planarity concepts of hypergraphs. For each embeddability class we give a full characterization. We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous embedding can be tested in linear time.

Keywords

Planar Graph Block Region Base Grid Full Embeddability Contour Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Athenstädt, J.C., Hartmann, T., Nöllenburg, M.: Simultaneous embeddability of two partitions. CoRR, abs/1408.6019 (August 2014)Google Scholar
  2. 2.
    Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, ch. 11, pp. 349–381. CRC Press (2013)Google Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Brandes, U., Cornelsen, S., Pampel, B., Sallaberry, A.: Path-based supports for hypergraphs. J. Discrete Algorithms 14, 248–261 (2012)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Buchin, K., van Kreveld, M., Meijer, H., Speckmann, B., Verbeek, K.: On planar supports for hypergraphs. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 345–356. Springer, Heidelberg (2010), see also Tech. Rep. UU-CS-2009-035, Utrecht University (2009) Google Scholar
  6. 6.
    Buja, A., Swayne, D.F., Littman, M.L., Dean, N., Hofmann, H., Chen, L.: Data visualization with multidimensional scaling. J. Comput. Graphical Statistics 17(2), 444–472 (2008)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Chaplick, S., Jelínek, V., Kratochvíl, J., Vyskočil, T.: Bend-bounded path intersection graphs: Sausages, noodles, and waffles on a grill. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) WG 2012. LNCS, vol. 7551, pp. 274–285. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Chow, S.: Generating and Drawing Area-Proportional Euler and Venn Diagrams. PhD thesis, University of Victoria (2007)Google Scholar
  9. 9.
    Collins, C., Penn, G., Carpendale, S.: Bubble sets: Revealing set relations with isocontours over existing visualizations. IEEE TVCG 15(6), 1009–1016 (2009)Google Scholar
  10. 10.
    de Berg, M., Khosravi, A.: Optimal binary space partitions in the plane. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 216–225. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Feng, Q.-W., Cohen, R., Eades, P.: Planarity for clustered graphs. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 213–226. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  12. 12.
    Flower, J., Fish, A., Howse, J.: Euler diagram generation. J. Visual Languages and Computing 19(6), 675–694 (2008)CrossRefGoogle Scholar
  13. 13.
    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)CrossRefMATHMathSciNetGoogle 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)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Kaufmann, M., van 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)CrossRefGoogle Scholar
  16. 16.
    Kohonen, T.: Self-Organizing Maps, 3rd edn. Springer (2001)Google Scholar
  17. 17.
    Mäkinen, E.: How to draw a hypergraph. Int. J. Computer Math. 34(3-4), 177–185 (1990)CrossRefMATHGoogle Scholar
  18. 18.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17(4), 717–728 (2001)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Simonetto, P., Auber, D., Archambault, D.: Fully automatic visualisation of overlapping sets. Computer Graphics Forum 28(3), 967–974 (2009)CrossRefGoogle Scholar
  20. 20.
    S. Wagner and D. Wagner. Comparing Clusterings – An Overview. Tech. Rep. 2006-04, Department of Informatics, Universität Karlsruhe, 2007.Google Scholar
  21. 21.
    Walsh, T.R.: Hypermaps Versus Bipartite Maps. J. Combinatorial Theory Series B 18(2), 155–163 (1975)CrossRefMATHGoogle Scholar
  22. 22.
    Zykov, A.A.: Hypergraphs. Russian Mathematical Surveys 29(6), 89–156 (1974)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jan Christoph Athenstädt
    • 1
  • Tanja Hartmann
    • 2
  • Martin Nöllenburg
    • 2
  1. 1.Department of Computer and Information ScienceUniversity of KonstanzGermany
  2. 2.Institute of Theoretical InformaticsKarlsruhe Institute of Technology (KIT)Germany

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