Proportional Contact Representations of 4-Connected Planar Graphs

  • Md. Jawaherul Alam
  • Stephen G. Kobourov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Abstract

In a contact representation of a planar graph, vertices are represented by interior-disjoint polygons and two polygons share a non-empty common boundary when the corresponding vertices are adjacent. In the weighted version, a weight is assigned to each vertex and a contact representation is called proportional if each polygon realizes an area proportional to the vertex weight. In this paper we study proportional contact representations of 4-connected internally triangulated planar graphs. The best known lower and upper bounds on the polygonal complexity for such graphs are 4 and 8, respectively. We narrow the gap between them by proving the existence of a representation with complexity 6. We then disprove a 10-year old conjecture on the existence of a Hamiltonian canonical cycle in a 4-connected maximal planar graph, which also implies that a previously suggested method for constructing proportional contact representations of complexity 6 for these graphs will not work. Finally we prove that it is NP-hard to decide whether a 4-connected planar graph admits a proportional contact representation using only rectangles.

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.
    Alam, M.J., Biedl, T.C., Felsner, S., Kaufmann, M., Kobourov, S.G., Ueckerdt, T.: Computing cartograms with optimal complexity. In: Symposium on Computational Geometry, SoCG 2012, pp. 21–30 (2012)Google Scholar
  3. 3.
    Buchsbaum, A.L., Gansner, E.R., Procopiuc, C.M., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Transactions on Algorithms 4(1) (2008)Google Scholar
  4. 4.
    Duncan, C.A., Gansner, E.R., Hu, Y.F., Kaufmann, M., Kobourov, S.G.: Optimal polygonal representation of planar graphs. Algorithmica 63(3), 672–691 (2012)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Eppstein, D., Mumford, E., Speckmann, B., Verbeek, K.: Area-universal and constrained rectangular layouts. SIAM Journal on Computing 41(3), 537–564 (2012)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Felsner, S., Francis, M.C.: Contact representations of planar graphs with cubes. In: Symposium on Computational Geometry, SoCG 2011, pp. 315–320 (2011)Google Scholar
  7. 7.
    de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: On triangle contact graphs. Combinatorics, Probability and Computing 3, 233–246 (1994)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    de Fraysseix, H., de Mendez, P.O.: On topological aspects of orientations. Discrete Mathematics 229(1-3), 57–72 (2001)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fusy, É.: Transversal structures on triangulations: A combinatorial study and straight-line drawings. Discrete Mathematics 309(7), 1870–1894 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    He, X.: On floor-plan of plane graphs. SIAM Journal on Computing 28(6), 2150–2167 (1999)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    House, D.H., Kocmoud, C.J.: Continuous cartogram construction. In: IEEE Visualization, pp. 197–204 (1998)Google Scholar
  12. 12.
    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
  13. 13.
    Koźmiński, K., Kinnen, E.: Rectangular duals of planar graphs. Networks 15, 145–157 (1985)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Liao, C.C., Lu, H.I., Yen, H.C.: Compact floor-planning via orderly spanning trees. Journal of Algorithms 48, 441–451 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Raisz, E.: The rectangular statistical cartogram. Geographical Review 24(3), 292–296 (1934)CrossRefGoogle Scholar
  16. 16.
    Sun, Y., Sarrafzadeh, M.: Floorplanning by graph dualization: L-shaped modules. Algorithmica 10(6), 429–456 (1993)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Tobler, W.: Thirty five years of computer cartograms. Annals of Association of American Geographers 94, 58–73 (2004)CrossRefGoogle Scholar
  18. 18.
    Ungar, P.: On diagrams representing graphs. Journal of London Mathematical Society 28, 336–342 (1953)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Yeap, K.H., Sarrafzadeh, M.: Floor-planning by graph dualization: 2-concave rectilinear modules. SIAM Journal on Computing 22, 500–526 (1993)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Md. Jawaherul Alam
    • 1
  • Stephen G. Kobourov
    • 1
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA

Personalised recommendations