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Blocks of Hypergraphs

Applied to Hypergraphs and Outerplanarity
  • Ulrik Brandes
  • Sabine Cornelsen
  • Barbara Pampel
  • Arnaud Sallaberry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)

Abstract

A support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a connected subgraph. We show how to test in polynomial time whether a given hypergraph has a cactus support, i.e. a support that is a tree of edges and cycles. While it is \(\mathcal N\mathcal P\)-complete to decide whether a hypergraph has a 2-outerplanar support, we show how to test in polynomial time whether a hypergraph that is closed under intersections and differences has an outerplanar or a planar support. In all cases our algorithms yield a construction of the required support if it exists. The algorithms are based on a new definition of biconnected components in hypergraphs.

Keywords

Bipartite Graph Planar Support Connected Subgraph Outer Face Hasse Diagram 
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.
    Beeri, C., Fagin, R., Maier, D., Yannakakis, M.: On the desirability of acyclic database schemes. Journal of the Association for Computing Mashinery 30(4), 479–513 (1983)CrossRefzbMATHGoogle Scholar
  2. 2.
    Berge, C.: Graphs and Hypergraphs. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  3. 3.
    Blackwell, A.F., Marriott, K., Shimojima, A. (eds.): Diagrams 2004. LNCS (LNAI), vol. 2980. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  4. 4.
    Booth, K.S., Lueker, G.S.: Testing for the consecutives ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences 13, 335–379 (1976)CrossRefzbMATHGoogle Scholar
  5. 5.
    Buchin, K., van Kreveld, M., Meijer, H., Speckmann, B., Verbeek, K.: On planar supports for hypergraphs. Technical Report UU-CS-2009-035, Department of Information and Computing Sciences, Utrecht University (2009)Google Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Chimani, M., Gutwenger, C.: Algorithms for the hypergraph and the minor crossing number problems. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 184–195. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Chow, S.C.: Generating and drawing area-proportional Euler and Venn diagrams. PhD thesis, University of Victoria, British Columbia Canada (2007)Google Scholar
  9. 9.
    Didimo, W., Giordano, F., Liotta, G.: Overlapping cluster planarity. Journal on Graph Algorithms and Applications 12(3), 267–291 (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Dinitz, Y., Karzanov, A.V., Lomonosov, M.: On the structure of a family of minimal weighted cuts in a graph. In: Fridman, A. (ed.) Studies in Discrete Optimization, pp. 290–306. Nauka (1976) (in Russian)Google Scholar
  11. 11.
    Eschbach, T., Günther, W., Becker, B.: Orthogonal hypergraph drawing for improved visibility. Journal on Graph Algorithms and Applications 10(2), 141–157 (2006)CrossRefzbMATHGoogle Scholar
  12. 12.
    Flower, J., Fish, A., Howse, J.: Euler diagram generation. Journal on Visual Languages and Computing 19(6), 675–694 (2008)CrossRefGoogle Scholar
  13. 13.
    Hopcroft, J.E., Tarjan, R.E.: Efficient planarity testing. Journal of the Association for Computing Mashinery 21, 549–568 (1974)CrossRefzbMATHGoogle Scholar
  14. 14.
    Johnson, D.S., Pollak, H.O.: Hypergraph planarity and the complexity of drawing Venn diagrams. Journal of Graph Theory 11(3), 309–325 (1987)CrossRefzbMATHGoogle 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.
    Korach, E., Stern, M.: The clustering matroid and the optimal clustering tree. Mathematical Programming, Series B 98, 385–414 (2003)CrossRefzbMATHGoogle Scholar
  17. 17.
    Král’, D., Kratochvíl, J., Voss, H.-J.: Mixed hypercacti. Discrete Mathematics 286, 99–113 (2004)CrossRefzbMATHGoogle Scholar
  18. 18.
    Mäkinen, E.: How to draw a hypergraph. International Journal of Computer Mathematics 34, 177–185 (1990)CrossRefzbMATHGoogle Scholar
  19. 19.
    Mutton, P., Rodgers, P., Flower, J.: Drawing graphs in Euler diagrams. In: Blackwell, et al. (eds.) [13], pp. 66–81Google Scholar
  20. 20.
    Sander, G.: Layout of directed hypergraphs with orthogonal hyperedges. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 381–386. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  21. 21.
    Schaefer, M., Štefankovič, D.: Decidability of string graphs. Journal of Computer and System Sciences 68(2), 319–334 (2004)CrossRefzbMATHGoogle Scholar
  22. 22.
    Simonetto, P., Auber, D.: Visualise undrawable Euler diagrams. In: Proceedings of the 12th International Conference on Information Visualization (InfoVis 2008), pp. 594–599. IEEE Computer Society Press, Los Alamitos (2008)Google Scholar
  23. 23.
    Simonetto, P., Auber, D.: An heuristic for the construction of intersection graphs. In: Proceedings of the 13th International Conference on Information Visualization (InfoVis 2009), pp. 673–678. IEEE Computer Society Press, Los Alamitos (2009)Google Scholar
  24. 24.
    Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on Computing 13(3), 566–579 (1984)CrossRefzbMATHGoogle Scholar
  25. 25.
    van Cleemput, W.M.: On the planarity of hypergraphs. Proceedings of the IEEE 66(4), 514–515 (1978)CrossRefGoogle Scholar
  26. 26.
    Verroust, A., Viaud, M.-L.: Ensuring the drawability of extended Euler diagrams for up to 8 sets. In: Blackwell, et al. (eds.) [3], pp. 128–141Google Scholar
  27. 27.
    Walsh, T.R.S.: Hypermaps versus bipartite maps. Journal of Combinatorial Theory, Series B 18, 155–163 (1975)CrossRefzbMATHGoogle Scholar
  28. 28.
    Zykov, A.A.: Hypergraphs. Uspekhi Matematicheskikh Nauk 6, 89–154 (1974)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ulrik Brandes
    • 1
  • Sabine Cornelsen
    • 1
  • Barbara Pampel
    • 1
  • Arnaud Sallaberry
    • 2
  1. 1.Fachbereich Informatik & InformationswissenschaftUniversität KonstanzGermany
  2. 2.CNRS UMR 5800 LaBRIINRIA Bordeaux - Sud OuestPikkoFrance

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