Drawing Planar Graphs with Circular Arcs

  • C. C. Cheng
  • C. A. Duncan
  • M. T. Goodrich
  • S. G. Kobourov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1731)


In this paper we address the problem of drawing planar graphs with circular arcs while maintaining good angular resolution and small drawing area. We present a lower bound on the area of drawings in which edges are drawn using exactly one circular arc. We also give an algorithm for drawing n-vertex planar graphs such that the edges are sequences of two continuous circular arcs. The algorithm runs in O(n) time and embeds the graph on the O(n) × O(n) grid, while maintaining θ(1/d(v)) angular resolution, where d(v) is the degree of vertex v. Since in this case we use circular arcs of infinite radius, this is also the first algorithm to simultaneously achieve good angular resolution, small area and at most one bend per edge using straight-line segments. Finally, we show how to create drawings in which edges are smooth C1-continuous curves, represented by a sequence of at most three circular arcs.


  1. 1.
    G. Di Battista, P. Eades, R. Tamassia, and I. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Englewood Cliffs, NJ, 1999.MATHGoogle Scholar
  2. 2.
    M. Chrobak and T. Payne. A linear-time algorithm for drawing planar graphs. Inform. Process. Lett.,54:241–246, 1995.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    H. de Fraysseix, J. Pach, and R. Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41–51, 1990.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    I. Fary. On straight lines representation of planar graphs. Acta Sci. Math. Szeged, 11:229–233, 1948.MathSciNetGoogle Scholar
  5. 5.
    M. Formann, T. Hagerup, J. Haralambides, M. Kaufmann, F. T. Leighton, A. Simvonis, Emo Welzl, and G. Woeginger. Drawing graphs in the plane with high resolution. SIAM J. Comput., 22:1035–1052, 1993.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd Annu. European Sympos. Algorithms, volume 855 of Lecture Notes Comput. Sci., pages 12–23. Springer-Verlag, 1994.MathSciNetGoogle Scholar
  7. 7.
    M. T. Goodrich and C. G. Wagner. A framework for drawing planar graphs with curves and polylines. In Graph Drawing’ 98, pages 153–166, 1998.Google Scholar
  8. 8.
    C. Gutwenger and P. Mutzel. Planar polyline drawings with good angular resolution. In Graph Drawing’ 98, pages 167–182, 1998.Google Scholar
  9. 9.
    G. Kant. Drawing planar graphs using the lmc-ordering. In Proc. 33th Annu. IEEE Sympos. Found. Comput. Sci., pages 101–110, 1992.Google Scholar
  10. 10.
    G. Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16:4–32, 1996. (special issue on Graph Drawing, edited by G. Di Battista and R. Tamassia).Google Scholar
  11. 11.
    S. Malitz and A. Papakostas. On the angular resolution of planar graphs. SIAM J. Discrete Math., 7:172–183, 1994.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    W. Schnyder. Embedding planar graphs on the grid. In Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pages 138–148, 1990.Google Scholar
  13. 13.
    W. T. Tutte. How to draw a graph. Proceedings London Mathematical Society, 13(52):743–768, 1963.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    K. Wagner. Bemerkungen zum vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung, 46:26–32, 1936.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • C. C. Cheng
    • 1
  • C. A. Duncan
    • 1
  • M. T. Goodrich
    • 1
  • S. G. Kobourov
    • 1
  1. 1.The Johns Hopkins UniversityBaltimore

Personalised recommendations