Planar Graphs as VPG-Graphs

  • Steven Chaplick
  • Torsten Ueckerdt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Abstract

A graph is Bk-VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show that the 4-connected planar graphs are a subclass of the intersection graphs of Z-shapes (i.e., a special case of B2-VPG). Additionally, we demonstrate that a B2-VPG representation of a planar graph can be constructed in O(n3/2) time. We further show that the triangle-free planar graphs are contact graphs of: L-shapes, Γ-shapes, vertical segments, and horizontal segments (i.e., a special case of contact B1-VPG). From this proof we gain a new proof that bipartite planar graphs are a subclass of 2-DIR.

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References

  1. 1.
    Alam, M.J., Biedl, T., Felsner, S., Kaufmann, M., Kobourov, S.G.: Proportional Contact Representations of Planar Graphs. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 26–38. Springer, Heidelberg (2012)Google Scholar
  2. 2.
    Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: Vertex intersection graphs of paths on a grid. J. Graph Algorithms Appl. 16(2), 129–150 (2012)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Arroyo Hartman, I., Newman, I., Ziv, R.: On grid intersection graphs. Discrete Math. 87(1), 41–52 (1991)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    de Castro, N., Cobos, F.J., Dana, J.C., Márquez, A., Noy, M.: Triangle-free planar graphs and segment intersection graphs. J. Graph Algorithms Appl. 6(1), 7–26 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chalopin, J., Gonçalves, D., Ochem, P.: Planar graphs are in 1-STRING. In: 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 609–617 (2007)Google Scholar
  6. 6.
    Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane: extended abstract. In: 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 631–638 (2009)Google Scholar
  7. 7.
    Ehrlich, G., Even, S., Tarjan, R.: Intersection graphs of curves in the plane. J. Comb. Theory, Ser. B 21(1), 8–20 (1976)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Eppstein, D.: Regular labelings and geometric structures. In: 22nd Canadian Conference on Computational Geometry, CCCG 2010, pp. 125–130 (2010)Google Scholar
  9. 9.
    de Fraysseix, H., Ossona de Mendez, P.: Representations by contact and intersection of segments. Algorithmica 47(4), 453–463 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    de Fraysseix, H., Ossona de Mendez, P., Pach, J.: Representation of planar graphs by segments. Intuitive Geometry 63, 109–117 (1991)Google Scholar
  11. 11.
    de Fraysseix, H., Ossona de Mendez, P., Rosenstiehl, P.: On triangle contact graphs. Combinatorics, Probability & Computing 3, 233–246 (1994)MATHCrossRefGoogle Scholar
  12. 12.
    Fusy, E.: Combinatoire des cartes planaires et applications algorithmiques. PhD Thesis (2007)Google Scholar
  13. 13.
    Fusy, E.: Transversal structures on triangulations: A combinatorial study and straight-line drawings. Discrete Math. 309(7), 1870–1894 (2009)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Gavenčiak, T.: Private communication: Small planar graphs are L-graphs (2012)Google Scholar
  15. 15.
    He, X.: On finding the rectangular duals of planar triangular graphs. SIAM J. Comput. 22, 1218–1226 (1993)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Heldt, D., Knauer, K., Ueckerdt, T.: On the Bend-Number of Planar and Outerplanar Graphs. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 458–469. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  17. 17.
    Kant, G., He, X.: Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theor. Comput. Sci. 172(1-2), 175–193 (1997)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kobourov, S., Ueckerdt, T., Verbeek, K.: Combinatorial and geometric properties of laman graphs (2012)Google Scholar
  19. 19.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig. Math.-Phys. Klasse 88, 141–164 (1936)Google Scholar
  20. 20.
    Koźmiński, K., Kinnen, E.: Rectangular dual of planar graphs. Networks 15(2), 145–157 (1985)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Comb. Theory, Ser. B 62(2), 289–315 (1994)MATHCrossRefGoogle Scholar
  22. 22.
    Scheinerman, E.R.: Intersection Classes and Multiple Intersection Parameters of Graphs. Ph.D. thesis. Princeton University (1984)Google Scholar
  23. 23.
    Sinden, F.: Topology of thin film circuits. Bell System Tech. J. 45, 1639–1662 (1966)MATHGoogle Scholar
  24. 24.
    Sun, Y., Sarrafzadeh, M.: Floorplanning by graph dualization: L-shaped modules. Algorithmica 10, 429–456 (1993)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Thomassen, C.: Interval representations of planar graphs. J. Comb. Theory, Ser. B 40(1), 9–20 (1986)MATHCrossRefGoogle Scholar
  26. 26.
    Ungar, P.: On diagrams representing graphs. J. London Math. Soc. 28, 336–342 (1953)Google Scholar
  27. 27.
    West, D.: Open problems. SIAM J. Discrete Math. Newslett. 2, 10–12 (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Steven Chaplick
    • 1
  • Torsten Ueckerdt
    • 2
  1. 1.School of Computing ScienceSimon Fraser UniversityCanada
  2. 2.Department of Applied MathematicsCharles UniversityCzech Republic

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