Orthogeodesic Point-Set Embedding of Trees

  • Emilio Di Giacomo
  • Fabrizio Frati
  • Radoslav Fulek
  • Luca Grilli
  • Marcus Krug
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7034)

Abstract

Let S be a set of N grid points in the plane, and let G a graph with n vertices (n ≤ N). An orthogeodesic point-set embedding of G on S is a drawing of G such that each vertex is drawn as a point of S and each edge is an orthogonal chain with bends on grid points whose length is equal to the Manhattan distance. We study the following problem. Given a family of trees \(\mathcal F\) what is the minimum value f(n) such that every n-vertex tree in \(\mathcal F\) admits an orthogeodesic point-set embedding on every grid-point set of size f(n)? We provide polynomial upper bounds on f(n) for both planar and non-planar orthogeodesic point-set embeddings as well as for the case when edges are required to be L-shaped chains.

References

  1. 1.
    Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. Theoretical Computer Science 408(2-3), 129–142 (2008)MathSciNetCrossRefMATHGoogle 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. 36(2), 117–130 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bose, P.: On embedding an outer-planar graph on a point set. Computational Geometry: Theory and Applications 23, 303–312 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brandenburg, F.J.: Drawing planar graphs on \(\frac{8}{9}n^2\) area. Electronic Notes in Discrete Mathematics 31, 37–40 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cabello, S.: Planar embeddability of the vertices of a graph using a fixed point set is NP-hard. Journal of Graph Algorithms and Applications 10(2), 353–366 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Di Giacomo, E., Grilli, L., Krug, M., Liotta, G., Rutter, I.: Hamiltonian Orthogeodesic Alternating Paths. In: Iliopoulos, C.S. (ed.) IWOCA 2011. LNCS, vol. 7056, pp. 170–181. Springer, Heidelberg (2011)Google Scholar
  8. 8.
    Di Giacomo, E., Liotta, G., Trotta, F.: Drawing colored graphs with constrained vertex positions and few bends per edge. Algorithmica 57, 796–818 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Mathematica 2, 463–470 (1935)MathSciNetMATHGoogle Scholar
  10. 10.
    Everett, H., Lazard, S., Liotta, G., Wismath, S.: Universal sets of n points for one-bend drawings of planar graphs with n vertices. Discrete and Computational Geometry 43, 272–288 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fink, M., Haunert, J.-H., Mchedlidze, T., Spoerhase, J., Wolff, A.: Drawing graphs with vertices at specified positions and crossings at large angles. pre-print, arXiv:1107.4970v1 (July 2011)Google Scholar
  12. 12.
    Di Giacomo, E., Frati, F., Fulek, R., Grilli, L., Krug, M.: Orthogeodesic point-set embedding of trees. Technical Report 2011-24, Kalrsruhe Institute of Technology, KIT (2011)Google Scholar
  13. 13.
    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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Katz, B., Krug, M., Rutter, I., Wolff, A.: Manhattan-Geodesic Embedding of Planar Graphs. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 207–218. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kurowski, M.: A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs. Information Processing Letters 92(2), 95–98 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Emilio Di Giacomo
    • 1
  • Fabrizio Frati
    • 2
    • 3
  • Radoslav Fulek
    • 2
  • Luca Grilli
    • 1
  • Marcus Krug
    • 4
  1. 1.Dip. di Ingegneria Elettronica e dell’InformazioneUniversitá degli Studi di PerugiaItaly
  2. 2.School of Basic SciencesÉcole Polytechnique Fédérale de LausanneSwitzerland
  3. 3.School of Information TechnologiesUniversity of SydneyAustralia
  4. 4.Institute of Theoretical InformaticsKarlsruhe Institute of TechnologyGermany

Personalised recommendations