Linear Size Universal Point Sets for Classes of Planar Graphs

Authors Stefan Felsner , Hendrik Schrezenmaier , Felix Schröder , Raphael Steiner

Thumbnail PDF


  • Filesize: 0.83 MB
  • 16 pages

Document Identifiers

Author Details

Stefan Felsner
  • Institute of Mathematics, Technische Universität Berlin, Germany
Hendrik Schrezenmaier
  • Institute of Mathematics, Technische Universität Berlin, Germany
Felix Schröder
  • Institute of Mathematics, Technische Universität Berlin, Germany
Raphael Steiner
  • Institute of Theoretical Computer Science, Department of Computer Science, ETH Zürich, Switzerland


We are highly indebted to Henry Förster, Linda Kleist, Joachim Orthaber and Marco Ricci due to discussions during GG-Week 2022 resulting in a solution to the problem of separating 2-cycles in our proof for subcubic graphs.

Cite AsGet BibTex

Stefan Felsner, Hendrik Schrezenmaier, Felix Schröder, and Raphael Steiner. Linear Size Universal Point Sets for Classes of Planar Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


A finite set P of points in the plane is n-universal with respect to a class 𝒞 of planar graphs if every n-vertex graph in 𝒞 admits a crossing-free straight-line drawing with vertices at points of P. For the class of all planar graphs the best known upper bound on the size of a universal point set is quadratic and the best known lower bound is linear in n. Some classes of planar graphs are known to admit universal point sets of near linear size, however, there are no truly linear bounds for interesting classes beyond outerplanar graphs. In this paper, we show that there is a universal point set of size 2n-2 for the class of bipartite planar graphs with n vertices. The same point set is also universal for the class of n-vertex planar graphs of maximum degree 3. The point set used for the results is what we call an exploding double chain, and we prove that this point set allows planar straight-line embeddings of many more planar graphs, namely of all subgraphs of planar graphs admitting a one-sided Hamiltonian cycle. The result for bipartite graphs also implies that every n-vertex plane graph has a 1-bend drawing all whose bends and vertices are contained in a specific point set of size 4n-6, this improves a bound of 6n-10 for the same problem by Löffler and Tóth.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graphs and surfaces
  • Human-centered computing → Graph drawings
  • Theory of computation → Computational geometry
  • Theory of computation → Randomness, geometry and discrete structures
  • Graph drawing
  • Universal point set
  • One-sided Hamiltonian
  • 2-page book embedding
  • Separating decomposition
  • Quadrangulation
  • 2-tree
  • Subcubic planar graph


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. A. Alahmadi, R.E.L. Aldred, and C. Thomassen. Cycles in 5-connected triangulations. Journal of Combinatorial Theory, Series B, 140:27-44, 2020. URL:
  2. Md. Jawaherul Alam, Therese C. Biedl, Stefan Felsner, Michael Kaufmann, Stephen G. Kobourov, and Torsten Ueckerdt. Computing cartograms with optimal complexity. Discret. Comput. Geom., 50:784-810, 2013. Google Scholar
  3. Md. Jawaherul Alam and Stephen G. Kobourov. Proportional contact representations of 4-connected planar graphs. In Graph Drawing, volume 7704 of LNCS, pages 211-223. Springer, 2012. Google Scholar
  4. Patrizio Angelini, Till Bruckdorfer, Giuseppe Di Battista, Michael Kaufmann, Tamara Mchedlidze, Vincenzo Roselli, and Claudio Squarcella. Small universal point sets for k-outerplanar graphs. Discrete & Computational Geometry, pages 1-41, 2018. URL:
  5. Michael J. Bannister, Zhanpeng Cheng, William E. Devanny, and David Eppstein. Superpatterns and Universal Point Sets. Journal of Graph Algorithms and Applications, 18(2):177-209, 2014. URL:
  6. Jean Cardinal, Michael Hoffmann, and Vincent Kusters. On Universal Point Sets for Planar Graphs. Journal of Graph Algorithms and Applications, 19(1):529-547, 2015. URL:
  7. Alexander Choi, Marek Chrobak, and Kevin Costello. An Ω(n²) lower bound for random universal sets for planar graphs. arXiv preprint, arXiv1908.07097, 2019. URL:
  8. Marek Chrobak and Howard J. Karloff. A Lower Bound on the Size of Universal Sets for Planar Graphs. ACM SIGACT News, 20(4):83-86, 1989. URL:
  9. Hubert De Fraysseix, János Pach, and Richard Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41-51, 1990. URL:
  10. Stefan Felsner, Éric Fusy, Marc Noy, and David Orden. Bijections for Baxter families and related objects. Journal of Combinatorial Theory, Series A, 118(3):993-1020, 2011. URL:
  11. Stefan Felsner, Clemens Huemer, Sarah Kappes, and David Orden. Binary labelings for plane quadrangulations and their relatives. Discrete Mathematics and Theoretical Computer Science, 12:3:115-138, 2010. Google Scholar
  12. Radoslav Fulek and Csaba D. Tóth. Universal point sets for planar three-trees. Journal of Discrete Algorithms, 30:101-112, 2015. URL:
  13. Peter Gritzmann, Bojan Mohar, János Pach, and Richard Pollack. Embedding a planar triangulation with vertices at specified points. American Mathematical Monthly, 98:165-166, 1991. URL:
  14. Michael Kaufmann and Roland Wiese. Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications, 6(1):115-129, 2002. URL:
  15. Maciej Kurowski. A 1.235n lower bound on the number of points needed to draw all n-vertex planar graphs. Information Processing Letters, 92(2):95-98, 2004. URL:
  16. Maarten Löffler and Csaba D. Tóth. Linear-size universal point sets for one-bend drawings. In Graph Drawing, volume 9411 of LNCS, pages 423-429. Springer, 2015. URL:
  17. Bojan Mohar. Universal point sets for planar graphs. Open Problem Garden, 2007. URL:
  18. Patrice Ossona de Mendez and Hubert de Fraysseix. On topological aspects of orientations. Discrete Mathematics, 229(1-3):57-72, 2001. URL:
  19. Julius Petersen. Die Theorie der regulären graphs. Acta Mathematica, 15(1):193-221, 1891. URL:
  20. Manfred Scheucher, Hendrik Schrezenmaier, and Raphael Steiner. A note on universal point sets for planar graphs. Journal of Graph Algorithms and Applications, 24(3):247-267, 2020. URL:
  21. Walter Schnyder. Embedding Planar Graphs on the Grid. In Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 138-148. Society for Industrial and Applied Mathematics, 1990. Google Scholar