Representation of Planar Hypergraphs by Contacts of Triangles

  • Hubert de Fraysseix
  • Patrice Ossona de Mendez
  • Pierre Rosenstiehl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)

Abstract

Many representation theorems extend from planar graphs to planar hypergraphs. The authors proved in [10] that every planar graph has a representation by contact of triangles. We prove here that this representation result extend to planar linear hypergraphs. Although the graph proof was simple and led to a linear time drawing algorithm, the extension for hypergraphs needs more work. The proof we give here relies on a combinatorial characterization of those hypergraphs which are representable by contact of segments in the plane, We propose some possible generalization directions and open problems, related to the order dimension of the incidence posets of hypergraphs.

References

  1. 1.
    Andreev, E.M.: On convex polyhedra in Lobačevskiǐ spaces. Matematicheskii Sbornik 81, 445–478 (1970)Google Scholar
  2. 2.
    Berge, C.: Graphes et hypergraphes, 2nd edn. Dunod, Paris (1973)MATHGoogle Scholar
  3. 3.
    Brightwell, G., Trotter, W.T.: The order dimension of planar maps. SIAM journal on Discrete Mathematics 10(4), 515–528 (1997)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cori, R.: Un code pour les graphes planaires et ses applications, Société Mathématique de France, Paris, vol. 27 (1975)Google Scholar
  5. 5.
    Cori, R., Machì, A.: Maps, hypermaps and their automorphisms. Expo. Math. 10, 403–467 (1992)MATHGoogle Scholar
  6. 6.
    Dushnik, B., Miller, E.W.: Partially ordered sets. Amer. J. Math. 63, 600–610 (1941)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    de Fraysseix, H., Ossona de Mendez, P.: Intersection Graphs of Jordan Arcs, Contemporary Trends in Discrete Mathematics. In: DIMATIA-DIMACS. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Štiřin 1997 Proc., pp. 11–28 (1999)Google Scholar
  8. 8.
    de Fraysseix, H., Ossona de Mendez, P.: Barycentric systems and stretchability. Discrete Applied Mathematics 155(9), 1079–1095 (2007)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    de Fraysseix, H., Ossona de Mendez, P.: On representations by contact and intersection of segments. Algorithmica 47(4), 453–463 (2007)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    de Fraysseix, H., Ossona de Mendez, P., Rosenstiehl, P.: On triangle contact graphs. Combinatorics, Probability and Computing 3, 233–246 (1994)MATHMathSciNetGoogle Scholar
  11. 11.
    de Fraysseix, H., Pach, J., Pollack, R.: Small sets supporting Fary embeddings of planar graphs. In: 20th Annual ACM Symposium on Theory of Computing, pp. 426–433 (1988)Google Scholar
  12. 12.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Johnson, D.S., Pollak, H.O.: Hypergraph planarity and the complexity of drawing Venn diagrams. Journal of Graph Theory 11(3), 309–325 (1987)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jones, R.P.: Colourings of hypergraphs, Ph.D. thesis, Royal Holloway College, Egham, p. 209 (1976)Google Scholar
  15. 15.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Verh. Schs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88, 141–164 (1936)Google Scholar
  16. 16.
    Ossona de Mendez, P.: Orientations bipolaires, Ph.D. thesis, Ecole des Hautes Etudes en Sciences Sociales, Paris (1994)Google Scholar
  17. 17.
    Ossona de Mendez, P.: Geometric Realization of Simplicial Complexes, Graph Drawing. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 323–332. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  18. 18.
    Ossona de Mendez, P.: Realization of posets. Journal of Graph Algorithms and Applications 6(1), 149–153 (2002)MATHMathSciNetGoogle Scholar
  19. 19.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layout and bipolar orientation of planar graphs. Discrete and Computational Geometry 1, 343–353 (1986)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Scheinerman, E.R.: Intersection classes and multiple intersection parameters of graphs, Ph.D. thesis, Princeton University (1984)Google Scholar
  21. 21.
    Scheinerman, E.R., West, D.B.: The interval number of a planar graph: Three intervals suffice. Journal of Combinatorial Theory, Series B 35, 224–239 (1983)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Schnyder, W.: Embedding planar graphs in the grid. In: First ACM-SIAM Symposium on Discrete Algorithms, pp. 138–147 (1990)Google Scholar
  24. 24.
    Tamassia, R., Tollis, I.G.: Tessalation representation of planar graphs. In: Proc. Twenty-Seventh Annual Allerton Conference on Communication, Control, and Computing, pp. 48–57 (1989)Google Scholar
  25. 25.
    Trotter, W.T.: Combinatorics and partially ordered sets: Dimension theory. John Hopkins series in the mathematical sciences. Johns Hopkins University Press, London (1992)MATHGoogle Scholar
  26. 26.
    Walsh, T.R.S.: Hypermaps versus bipartite maps. J. Combinatorial Theory 18(B), 155–163 (1975)MATHGoogle Scholar
  27. 27.
    White, A.T.: Graphs, Groups and Surfaces, revised edn. Mathematics Studies, vol. 8. North-Holland, Amsterdam (1984)Google Scholar
  28. 28.
    Zykov, A.A.: Hypergraphs. Uspeki Mat. Nauk 6, 89–154 (1974)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Hubert de Fraysseix
    • 1
  • Patrice Ossona de Mendez
    • 1
  • Pierre Rosenstiehl
    • 1
  1. 1.CNRS/EHESSCAMS UMR 8557ParisFrance

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