Testing Maximal 1-Planarity of Graphs with a Rotation System in Linear Time

(Extended Abstract)
  • Peter Eades
  • Seok-Hee Hong
  • Naoki Katoh
  • Giuseppe Liotta
  • Pascal Schweitzer
  • Yusuke Suzuki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)


A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. It is known that testing 1-planarity of a graph is NP-complete. A 1-planar embedding of a graph G is maximal if no edge can be added without violating the 1-planarity of G. In this paper we show that the problem of testing maximal 1-planarity of a graph G can be solved in linear time, if a rotation system (i.e., the circular ordering of edges for each vertex) is given. We also prove that there is at most one maximal 1-planar embedding of G that preserves the given rotation system, and our algorithm produces such an embedding in linear time, if it exists.


Linear Time Rotation System Linear Time Algorithm Blue Edge Embed Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Auer, C., Brandenburg, F.J., Gleißner, A., Reislhuber, J.: On 1-planar graphs with rotation systems. Tech. Rep. MIP1207, Faculty of Informatics and Mathematics, University of Passau (2012)Google Scholar
  2. 2.
    Borodin, O.V.: Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs. Metody Diskret. Analiz. 41, 12–26, 108 (1984)Google Scholar
  3. 3.
    Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. CoRR abs/1203.5944 (2012)Google Scholar
  4. 4.
    Eades, P., Hong, S.H., Katoh, N., Liotta, G., Schweitzer, P., Suzuki, Y.: Testing maximal 1-planarity of graphs with a rotation system in linear time. TR IT-IVG-2012-02, School of IT, University of Sydney (2012)Google Scholar
  5. 5.
    Eades, P., Liotta, G.: Right Angle Crossing Graphs and 1-Planarity. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 148–153. Springer, Heidelberg (2012)Google Scholar
  6. 6.
    Hong, S.-H., Eades, P., Liotta, G., Poon, S.-H.: Fáry’s Theorem for 1-Planar Graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 335–346. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Hudák, D., Madaras, T.: On local properties of 1-planar graphs with high minimum degree. Ars Math. Contemp. 4(2), 245–254 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. J. Graph Theory (2012)Google Scholar
  9. 9.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ringel, G.: Ein Sechsfarbenproblem auf der Kugel. Abh. Math. Sem. Univ. Hamburg 29, 107–117 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Suzuki, Y.: Optimal 1-planar graphs which triangulate other surfaces. Discrete Mathematics 310(1), 6–11 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Suzuki, Y.: Re-embeddings of maximum 1-planar graphs. SIAM J. Discrete Math. 24(4), 1527–1540 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Thomassen, C.: Rectilinear drawings of graphs. Journal of Graph Theory 12(3), 335–341 (1988)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Peter Eades
    • 1
  • Seok-Hee Hong
    • 1
  • Naoki Katoh
    • 2
  • Giuseppe Liotta
    • 3
  • Pascal Schweitzer
    • 4
  • Yusuke Suzuki
    • 5
  1. 1.University of SydneyAustralia
  2. 2.Kyoto UniversityJapan
  3. 3.Universitá di PerugiaItaly
  4. 4.Australian National UniversityAustralia
  5. 5.Niigata UniversityJapan

Personalised recommendations