Discrete & Computational Geometry

, Volume 43, Issue 2, pp 272–288

Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices

  • Hazel Everett
  • Sylvain Lazard
  • Giuseppe Liotta
  • Stephen Wismath
Article

Abstract

This paper shows that any planar graph with n vertices can be point-set embedded with at most one bend per edge on a universal set of n points in the plane. An implication of this result is that any number of planar graphs admit a simultaneous embedding without mapping with at most one bend per edge.

Keywords

Graph drawing Universal point sets One-bend drawings Simultaneous embeddings 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bernhart, F., Kainen, P.C.: The book thickness of a graph. J. Comb. Theory, Ser. B 27, 320–331 (1979) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Braß, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geom. Theory Appl. 36(2), 117–130 (2007) MATHGoogle Scholar
  3. 3.
    Chrobak, M., Karloff, H.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20(4), 83–86 (1989) CrossRefGoogle Scholar
  4. 4.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Comput. Geom. Theory Appl. 30, 1–23 (2005) MATHMathSciNetGoogle Scholar
  6. 6.
    Enomoto, H., Miyauchi, M.S.: Embedding graphs into a three page book with o(mlog n) crossings of edges over the spine. SIAM J. Discrete Math. 12(3), 337–341 (1999) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Am. Math. Mon. 98(2), 165–166 (1991) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002) MATHMathSciNetGoogle Scholar
  9. 9.
    Kurowski, M.: A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs. Inf. Process. Lett. 92(2), 95–98 (2004) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs Comb. 17, 717–728 (2001) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM–SIAM Sympos. Discrete Algorithms (SODA’90), pp. 138–148 (1990) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Hazel Everett
    • 1
  • Sylvain Lazard
    • 1
  • Giuseppe Liotta
    • 2
  • Stephen Wismath
    • 3
  1. 1.INRIA Nancy Grand EstUniversité Nancy 2, LORIANancyFrance
  2. 2.Dip. di Ingegneria Elettronica e dell’InformazioneUniversità degli Studi di PerugiaPerugiaItaly
  3. 3.Department of Mathematics and Computer ScienceUniversity of LethbridgeLethbridgeCanada

Personalised recommendations