On Universal Point Sets for Planar Graphs

  • Jean Cardinal
  • Michael Hoffmann
  • Vincent Kusters
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8296)


A set P of points in ℝ2 is n-universal, if every planar graph on n vertices admits a plane straight-line embedding on P. Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n ≥ 15. Conversely, we use a computer program to show that there exist universal point sets for all n ≤ 10 and to enumerate all corresponding order types. Finally, we describe a collection \(\mathcal{G}\) of 7′393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane straight-line embedding of all graphs in \(\mathcal{G}\).


Planar Graph Outer Face Construction Sequence Facial Triangle Maximal Planar Graph 
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  1. 1.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pp. 138–148 (1990)Google Scholar
  3. 3.
    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
  4. 4.
    Chrobak, M., Karloff, H.J.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20(4), 83–86 (1989)CrossRefGoogle Scholar
  5. 5.
    Demaine, E.D., Mitchell, J.S.B., O’Rourke, J.: The Open Problems Project, Problem #45,
  6. 6.
    Kobourov, S.G.: Personal communication (2012)Google Scholar
  7. 7.
    Ábrego, B.M., Fernández-Merchant, S.: A lower bound for the rectilinear crossing number. Graphs and Combinatorics 21(3), 293–300 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lovász, L., Vesztergombi, K., Wagner, U., Welzl, E.: Convex quadrilaterals and k-sets. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 324, pp. 139–148. American Mathematical Society, Providence (2004)CrossRefGoogle Scholar
  9. 9.
    Fulek, R., Tóth, C.D.: Universal point sets for planar three-trees. CoRR abs/1212.6148 (2012)Google Scholar
  10. 10.
    Fáry, I.: On straight lines representation of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)zbMATHGoogle Scholar
  11. 11.
    Wagner, K.: Bemerkungen zum Vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung 46, 26–32 (1936)Google Scholar
  12. 12.
    Beineke, L.W., Pippert, R.E.: Enumerating dissectible polyhedra by their automorphism groups. Canad. J. Math. 26, 50–67 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brass, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D.P., Kobourov, S.G., Lubiw, A., Mitchell, J.S.: On simultaneous planar graph embeddings. Comput. Geom. Theory Appl. 36(2), 117–130 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Goodman, J.E., Pollack, R.: Multidimensional sorting. SIAM J. Comput. 12(3), 484–507 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Aichholzer, O., Krasser, H.: The point set order type data base: A collection of applications and results. In: Proc. 13th Canad. Conf. Comput. Geom., Waterloo, Canada, pp. 17–20 (2001)Google Scholar
  16. 16.
    Cardinal, J., Hoffmann, M., Kusters, V.: A program to find all universal point sets (2013),
  17. 17.
    Brinkmann, G., McKay, B.: Fast generation of planar graphs. MATCH Communications in Mathematical and in Computer Chemistry 58(2), 323–357 (2007)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Brinkmann, G., McKay, B.: The program plantri (2007),

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jean Cardinal
    • 1
  • Michael Hoffmann
    • 2
  • Vincent Kusters
    • 2
  1. 1.Département d’InformatiqueUniversité Libre de Bruxelles (ULB)Belgium
  2. 2.Institute of Theoretical Computer ScienceETH ZürichSwitzerland

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