Point-Set Embedding of Trees with Edge Constraints

(Extended Abstract)
  • Emilio Di Giacomo
  • Walter Didimo
  • Giuseppe Liotta
  • Henk Meijer
  • Stephen Wismath
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)

Abstract

Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph G′ ⊂ G on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of G′. We concentrate on trees and show how to compute the output in O(n2 logn) time and with at most 1 + 2 ⌈k/2 ⌉ bends per edge, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least k − 3 bends for some of the edges.

References

  1. 1.
    Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. In: WADS 2007. LNCS, Springer, Heidelberg (2007)Google Scholar
  2. 2.
    Bose, P.: On embedding an outer-planar graph on a point set. Computational Geometry: Theory and Applications 23, 303–312 (2002)MATHMathSciNetGoogle Scholar
  3. 3.
    Bose, P., McAllister, M., Snoeyink, J.: Optimal algorithms to embed trees in a point set. Journal of Graph Algorithms and Applications 2(1), 1–15 (1997)MathSciNetGoogle Scholar
  4. 4.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice-Hall, Upper Saddle River (1999)MATHCrossRefGoogle Scholar
  5. 5.
    Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: k-colored point-set embeddability of outerplanar graphs. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 318–329. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Di Giacomo, E., Liotta, G., Trotta, F.: On embedding a graph on two sets of points. International Journal of Foundations of Computer Science, Special Issue on Graph Drawing 17(5), 1071–1094 (2006)MATHGoogle Scholar
  7. 7.
    Halton, J.H.: On the thickness of graphs of given degree. Information Sciences 54, 219–238 (1991)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ikebe, Y., Perles, M., Tamura, A., Tokunaga, S.: The rooted tree embedding problem into points in the plane. Discrete Comput. Geometry 11, 51–63 (1994)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kaneko, A., Kano, M.: Straight line embeddings of rooted star forests in the plane. Discrete Appl. Mathematics 101, 167–175 (2000)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kaneko, A., Kano, M.: Semi-balanced partitions of two sets of points and embeddings of rooted forests. Int. J. Comput. Geom. Appl. 15(3), 229–238 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001)MATHGoogle Scholar
  12. 12.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications 6(1), 115–129 (2002)MATHMathSciNetGoogle Scholar
  13. 13.
    Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. Lecture Notes Series on Computing, vol. 12. World Scientific, Singapore (2004)MATHGoogle Scholar
  14. 14.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford Univ. Press, Oxford (1987)MATHGoogle Scholar
  15. 15.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17, 717–728 (2001)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Patrignani, M.: On extending a partial straight-line drawing. International Journal of Foundations of Computer Science, Special Issue on Graph Drawing 17(5), 1061–1069 (2006)MATHMathSciNetGoogle Scholar
  17. 17.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction, 3rd edn. Springer, Heidelberg (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Emilio Di Giacomo
    • 1
  • Walter Didimo
    • 1
  • Giuseppe Liotta
    • 1
  • Henk Meijer
    • 2
  • Stephen Wismath
    • 3
  1. 1.Dip. di Ingegneria Elettronica e dell’InformazioneUniversità degli Studi di Perugia 
  2. 2.Roosevelt AcademyThe Netherlands
  3. 3.Department of Mathematics and Computer ScienceUniversity of Lethbridge 

Personalised recommendations