Visibility Representations of Four-Connected Plane Graphs with Near Optimal Heights

  • Chieh-Yu Chen
  • Ya-Fei Hung
  • Hsueh-I Lu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)


A visibility representation of a graph G is to represent the nodes of G with non-overlapping horizontal line segments such that the line segments representing any two distinct adjacent nodes are vertically visible to each other. If G is a plane graph, i.e., a planar graph equipped with a planar embedding, a visibility representation of G has the additional requirement of reflecting the given planar embedding of G. For the case that G is an n-node four-connected plane graph, we give an O(n)-time algorithm to produce a visibility representation of G with height at most \(\left\lceil\frac{n}{2}\right\rceil+2\left\lceil\sqrt{\frac{n-2}{2}}\right\rceil\). To ensure that the first-order term of the upper bound is optimal, we also show an n-node four-connected plane graph G, for infinite number of n, whose visibility representations require heights at least \(\frac{n}{2}\).


Plane Graph Visibility Representation External Boundary Node Label External Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Even, S., Tarjan, R.E.: Computing an st-numbering. Theoretical Computer Science 2(3), 339–344 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Fan, J.H., Lin, C.C., Lu, H.I., Yen, H.C.: Width-optimal visibility representations of plane graphs. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 160–171. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    He, X., Kao, M.Y., Lu, H.I.: Linear-time succinct encodings of planar graphs via canonical orderings. SIAM Journal on Discrete Mathematics 12(3), 317–325 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Kant, G.: A more compact visibility representation. International Journal Computational Geometry and Applications 7(3), 197–210 (1997)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Kant, G., He, X.: Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theoretical Computer Science 172, 175–193 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Lin, C.C., Lu, H.I., Sun, I.F.: Improved compact visibility representation of planar graph via Schnyder’s realizer. SIAM Journal on Discrete Mathematics 18(1), 19–29 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Otten, R.H.J.M., van Wijk, J.G.: Graph representations in interactive layout design. In: Proceedings of the IEEE International Symposium on Circuits and Systems, pp. 914–918 (1978)Google Scholar
  8. 8.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete and Computational Geometry 1, 343–353 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Tamassia, R., Tollis, I.G.: A unified approach to visibility representations of planar graphs. Discrete and Computational Geometry 1, 321–341 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    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
  11. 11.
    Zhang, H., He, X.: On visibility representation of plane graphs. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 477–488. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Zhang, H., He, X.: Canonical ordering trees and their applications in graph drawing. Discrete and Computational Geometry 33, 321–344 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Zhang, H., He, X.: Improved visibility representation of plane graphs. Computational Geometry 30(1), 29–39 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Zhang, H., He, X.: New theoretical bounds of visibility representation of plane graphs. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 425–430. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Zhang, H., He, X.: Nearly optimal visibility representations of plane graphs. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 407–418. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Zhang, H., He, X.: Optimal st-orientations for plane triangulations. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 296–305. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Zhang, H., He, X.: Nearly optimal visibility representations of plane triangulations. SIAM Journal on Discrete Mathematics 22(4), 1364–1380 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Chieh-Yu Chen
    • 1
  • Ya-Fei Hung
    • 2
  • Hsueh-I Lu
    • 1
    • 2
  1. 1.Department of Computer Science and Information EngineeringNational Taiwan UniversityTaiwan
  2. 2.Graduate Institute of Networking and MultimediaNational Taiwan UniversityTaipei 106Taiwan, ROC

Personalised recommendations