1-Planar Graphs have Constant Book Thickness

  • Michael A. Bekos
  • Till Bruckdorfer
  • Michael Kaufmann
  • Chrysanthi Raftopoulou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)


In a book embedding the vertices of a graph are placed on the “spine” of a book and the edges are assigned to “pages”, so that edges on the same page do not cross. In this paper, we prove that every 1-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages. To the best of our knowledge, the best non-trivial previous upper-bound was \(O(\sqrt{n})\), where n is the number of vertices of the graph.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alam, M. J., Brandenburg, F.J., Kobourov, S.G.: Straight-line grid drawings of 3-connected 1-planar graphs. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 83–94. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. 2.
    Bekos, M., Gronemann, M., Raftopoulou, C.: Two-page book embeddings of 4-planar graphs. In: STACS. LIPIcs, vol. 25, pp. 137–148. Schloss Dagstuhl (2014)Google Scholar
  3. 3.
    Bekos, M.A., Bruckdorfer, T., Kaufmann, M., Raftopoulou, C.N.: The book thickness of 1-planar graphs is constant. CoRR, abs/1503.04990 (2015)Google Scholar
  4. 4.
    Bernhart, F., Kainen, P.: The book thickness of a graph. Combinatorial Theory 27(3), 320–331 (1979)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bilski, T.: Embedding graphs in books: a survey. IEEE Proceedings Computers and Digital Techniques 139(2), 134–138 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bodendiek, R., Schumacher, H., Wagner, K.: Über 1-optimale graphen. Mathematische Nachrichten 117(1), 323–339 (1984)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM Journal on Computing 42(5), 1803–1829 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cornuéjols, G., Naddef, D., Pulleyblank, W.: Halin graphs and the travelling salesman problem. Mathematical Programming 26(3), 287–294 (1983)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dujmovic, V., Wood, D.: Graph treewidth and geometric thickness parameters. Discrete and Computational Geometry 37(4), 641–670 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Grigoriev, A., Bodlaender, H.: Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49(1), 1–11 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Heath, L.: Embedding planar graphs in seven pages. In: FOCS, pp. 74–83. IEEE (1984)Google Scholar
  12. 12.
    Heath, L.: Algorithms for Embedding Graphs in Books. PhD thesis, University of North Carolina (1985)Google Scholar
  13. 13.
    Kainen, P., Overbay, S.: Extension of a theorem of whitney. AML 20(7), 835–837 (2007)MathSciNetMATHGoogle Scholar
  14. 14.
    Korzhik, V., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. Journal of Graph Theory 72(1), 30–71 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Malitz, S.: Genus g graphs have pagenumber \(O(\sqrt{g})\). Journal of Algorithms 17(1), 85–109 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Malitz, S.: Graphs with E edges have pagenumber \(O(\sqrt{E})\). Journal of Algorithms 17(1), 71–84 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nesetril, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures, and Algorithms. Algorithms and Combinatorics, vol. 28. Springer, New York (2012)MATHGoogle Scholar
  18. 18.
    Nishizeki, T., Chiba, N.: Planar Graphs: Theory and Algorithms. In: Hamiltonian Cycles, ch. 10. Dover Books on Mathematics, pp. 171–184. Courier Dover Publications (2008)Google Scholar
  19. 19.
    Ollmann, T.: On the book thicknesses of various graphs. In: Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing. Congressus Numerantium, vol. VIII, p. 459 (1973)Google Scholar
  20. 20.
    Overbay, S.: Graphs with small book thickness. Missouri Journal of Mathematical Science 19(2), 121–130 (2007)MathSciNetMATHGoogle Scholar
  21. 21.
    Wigderson, A.: The complexity of the hamiltonian circuit problem for maximal planar graphs. Technical Report TR-298, EECS Department, Princeton University (1982)Google Scholar
  22. 22.
    Yannakakis, M.: Embedding planar graphs in four pages. Journal of Computer and System Science 38(1), 36–67 (1989)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Michael A. Bekos
    • 1
  • Till Bruckdorfer
    • 1
  • Michael Kaufmann
    • 1
  • Chrysanthi Raftopoulou
    • 2
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany
  2. 2.School of Applied Mathematics and Physical ScienceNTUAAthensGreece

Personalised recommendations