A Note on Universal Point Sets for Planar Graphs

  • Manfred ScheucherEmail author
  • Hendrik Schrezenmaier
  • Raphael Steiner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)


We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices. We first show that at least \((1.293-o(1))n\) points are required to find a straight-line drawing of each n-vertex planar graph (vertices are drawn as the given points); this improves the previous best constant 1.235 by Kurowski (2004).

Our second main result is based on exhaustive computer search: We show that no set of 11 points exists, on which all planar 11-vertex graphs can be simultaneously drawn plane straight-line. This strengthens the result by Cardinal, Hoffmann, and Kusters (2015), that all planar graphs on \(n \le 10\) vertices can be simultaneously drawn on particular n-universal sets of n points while there are no n-universal sets of size n for \(n \ge 15\). We also provide 49 planar 11-vertex graphs which cannot be simultaneously drawn on any set of 11 points. This, in fact, is another step towards a (negative) answer of the question, whether every two planar graphs can be drawn simultaneously – a question raised by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw, and Mitchell (2007).


Simultaneously embedded Stacked triangulation Order type Boolean satisfiability (SAT) Integer programming (IP) 


  1. 1.
    IBM ILOG CPLEX Optimization Studio (2018).
  2. 2.
    Aichholzer, O.: Enumerating Order Types for Small Point Sets with Applications.
  3. 3.
    Aichholzer, O., Aurenhammer, F., Krasser, H.: Enumerating order types for small point sets with applications. Order 19(3), 265–281 (2002). Scholar
  4. 4.
    Aichholzer, O., Krasser, H.: Abstract order type extension and new results on the rectilinear crossing number. Comput. Geom.: Theory Appl. 36(1), 2–15 (2006). Scholar
  5. 5.
    Angelini, P., Bruckdorfer, T., Di Battista, G., Kaufmann, M., Mchedlidze, T., Roselli, V., Squarcella, C.: Small universal point sets for k-outerplanar graphs. Discret. Comput. Geom. 1–41 (2018).
  6. 6.
    Balko, M., Fulek, R., Kynčl, J.: Crossing numbers and combinatorial characterization of monotone drawings of \(K_n\). Discret. Comput. Geom. 53(1), 107–143 (2015).
  7. 7.
    Bannister, M.J., Cheng, Z., Devanny, W.E., Eppstein, D.: Superpatterns and universal point sets. J. Graph Algorithms Appl. 18(2), 177–209 (2014). Scholar
  8. 8.
    Brandenburg, F.J.: Drawing planar graphs on \(\frac{8}{9}n^2\) area. Electron. Notes Discret. Math. 31, 37–40 (2008). Scholar
  9. 9.
    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. 36(2), 117–130 (2007). Scholar
  10. 10.
    Brinkmann, G., McKay, B.D.: Fast generation of some classes of planar graphs. Electron. Notes Discret. Math. 3, 28–31 (1999). Scholar
  11. 11.
    Cabello, S.: Planar embeddability of the vertices of a graph using a fixed point set is NP-hard. J. Graph Algorithms Appl. 10(2), 353–363 (2006). Scholar
  12. 12.
    Cardinal, J., Hoffmann, M., Kusters, V.: On universal point sets for planar graphs. J. Graph Algorithms Appl. 19(1), 529–547 (2015). Scholar
  13. 13.
    Castañeda, N., Urrutia, J.: Straight line embeddings of planar graphs on point sets. In: Proceedings of the 8th Canadian Conference on Computational Geometry (CCCG 1996), pp. 312–318 (1996).
  14. 14.
    Chrobak, M., Karloff, H.J.: A lower bound on the size of universal sets for planar graphs. ACM SIGACT News 20(4), 83–86 (1989). Scholar
  15. 15.
    De Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990). Scholar
  16. 16.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004). Scholar
  17. 17.
    Felsner, S., Goodman, J.E.: Pseudoline arrangements. In: Toth, C.D., O’Rourke, J., Goodman, J.E. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press (2018).
  18. 18.
    Felsner, S., Weil, H.: Sweeps, arrangements and signotopes. Discret. Appl. Math. 109(1), 67–94 (2001). Scholar
  19. 19.
    Fulek, R., Tóth, C.D.: Universal point sets for planar three-trees. J. Discret. Algorithms 30, 101–112 (2015). Scholar
  20. 20.
    Gurobi Optimization, LLC: Gurobi Optimizer (2018).
  21. 21.
    Krasser, H.: Order Types of Point Sets in the Plane. Ph.D. thesis, Institute for Theoretical Computer Science, Graz University of Technology, Austria (2003)Google Scholar
  22. 22.
    Kurowski, M.: A \(1.235n\) lower bound on the number of points needed to draw all \(n\)-vertex planar graphs. Inf. Process. Lett. 92(2), 95–98 (2004).
  23. 23.
    McKay, B.D., Piperno, A.: Practical graph isomorphism, II. J. Symb. Comput. 60, 94–112 (2014). Scholar
  24. 24.
    Pach, J., Gritzmann, P., Mohar, B., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Am. Math. Mon. 98, 165–166 (1991). Scholar
  25. 25.
    Scheucher, M.: Webpage: Source Codes and Data for Universal Point Sets.
  26. 26.
    Scheucher, M.: On Order Types, Projective Classes, and Realizations. Bachelor’s thesis, Graz University of Technology, Austria (2014).
  27. 27.
    Scheucher, M., Schrezenmaier, H., Steiner, R.: A Note On Universal Point Sets for Planar Graphs. arXiv:1811.06482 (2018)
  28. 28.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 138–148. Society for Industrial and Applied Mathematics (1990)Google Scholar
  29. 29.
    Stein, W.A., et al.: Sage Mathematics Software (Version 8.1). The Sage Development Team (2018).
  30. 30.
    Stein, W.A., et al.: Sage Reference Manual: Graph Theory (Release 8.1) (2018).

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Manfred Scheucher
    • 1
    Email author
  • Hendrik Schrezenmaier
    • 1
  • Raphael Steiner
    • 1
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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