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)


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.


Planar Graph General Point Maximum Degree Outerplanar Graph Leftmost Point 
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  1. 1.
    Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. Theoretical Computer Science 408(2-3), 129–142 (2008)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bose, P.: On embedding an outer-planar graph on a point set. Computational Geometry: Theory and Applications 23, 303–312 (2002)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Mathematica 2, 463–470 (1935)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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

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