Constrained Point-Set Embeddability of Planar Graphs

  • Emilio Di Giacomo
  • Walter Didimo
  • Giuseppe Liotta
  • Henk Meijer
  • Stephen Wismath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)

Abstract

This paper starts the investigation of a constrained version of the point-set embeddability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: (i) If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E ∖ E′ have at most 4 bends each. (ii) If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E ∖ E′ is at most 8. (iii) If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E ∖ E′ is O(2k).

References

  1. 1.
    Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. Theoret. Comput. Sci. 408(2-3), 129–142 (2008)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bose, P.: On embedding an outer-planar graph on a point set. Comput. Geom. Theory Appl. 23, 303–312 (2002)CrossRefMATHGoogle Scholar
  3. 3.
    Bose, P., McAllister, M., Snoeyink, J.: Optimal algorithms to embed trees in a point set. J. Graph Algorithms Appl. 2(1), 1–15 (1997)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: k-colored point-set embeddability of outerplanar graphs. J. Graph Algorithms Appl. 11(1), 29–49 (2008)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Di Giacomo, E., Liotta, G., Trotta, F.: On embedding a graph on two sets of points. Internat. J. Found. Comput. Sci. 17(5), 1071–1094 (2006)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Wismath, S.K.: Point-set embeddings of trees with edge constraints. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 113–124. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Comput. Geom. Theory Appl. 30(1), 1–23 (2005)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Di Giacomo, E., Liotta, G., Trotta, F.: Drawing colored graphs with constrained vertex positions and few bends per edge. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 315–326. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Amer. Math. Monthly 98(2), 165–166 (1991)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Halton, J.H.: On the thickness of graphs of given degree. Inform. Sci. 54, 219–238 (1991)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Ikebe, Y., Perles, M., Tamura, A., Tokunaga, S.: The rooted tree embedding problem into points in the plane. Discrete Comput. Geom. 11, 51–63 (1994)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Kaneko, A., Kano, M.: Straight line embeddings of rooted star forests in the plane. Discrete Appl. Math. 101, 167–175 (2000)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kaneko, A., Kano, M.: Semi-balanced partitions of two sets of points and embeddings of rooted forests. Internat. J. Comput. Geom. Appl. 15(3), 229–238 (2005)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Pach, J., Törőcsik, J.: Layout of rooted trees. DIMACS Series in Discrete Math. and Theoretical Comput. Sci. 9, 131–137 (1993)Google Scholar
  16. 16.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17, 717–728 (2001)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Sugiyama, K.: Graph Drawing and Applications for Software and Knowledge Engineers. World Scientific, Singapore (2002)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

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 PerugiaItaly
  2. 2.Roosevelt AcademyThe Netherlands
  3. 3.Department of Mathematics and Computer ScienceUniversity of LethbridgeCanada

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