Spherical-Rectangular Drawings

  • Mahdieh Hasheminezhad
  • S. Mehdi Hashemi
  • Brendan D. McKay
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5431)

Abstract

We extend the concept of rectangular drawing to drawings on a sphere using meridians and circles of latitude such that each face is bounded by at most two circles and at most two meridians. This is called spherical-rectangular drawing. Special cases include drawing on a cylinder, a cone, or a lattice of concentric circles on the plane. In this paper, we prove necessary and sufficient conditions for cubic planar graphs to have spherical-rectangular drawings, and show that one can find in linear time a spherical-rectangular drawing of a subcubic planar graph if it has one.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Di Battista, G., Tammasia, R.: On-line planarity testing. SIAM Journal on Computing 5, 956–997 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Hasheminezhad, M., Hashemi, M.S., Tahmasbi, M.: Ortho-radial drawings of graphs. Australasian Journal Combinatorics (to appear)Google Scholar
  3. 3.
    Kowalik, L.: Short cycles in planar graphs. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 284–296. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Miura, K., Haga, H., Nishizeki, T.: Inner rectangular drawings of plane graphs. International Journal of Computational Geometry and Applications 16, 249–270 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Rahman, M.S., Nakano, S., Nishizeki, T.: Rectangular grid drawing of plane graphs. Computational Geometry 10, 203–220 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Rahman, M.S., Nakano, S., Nishizeki, T.: Rectangular drawings of plane graphs without designated corners. Computational Geometry 21, 121–138 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Rahman, M.S., Nishizeki, T., Ghosh, S.: Rectangular drawings of planar graphs. Journal of Algorithms 50, 62–78 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Thomassen, C.: Plane representations of graphs. Progress in Graph Theory, 43–69 (1984)Google Scholar
  9. 9.
    Zhang, H., He, X.: Compact visibility representation and straight-line grid embedding of plane graphs. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 493–504. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Mahdieh Hasheminezhad
    • 1
  • S. Mehdi Hashemi
    • 1
  • Brendan D. McKay
    • 2
  1. 1.Department of Computer Science, Faculty of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran
  2. 2.Department of Computer ScienceAustralian National UniversityCanberraAustralia

Personalised recommendations