A Linear-Time Algorithm for Testing Outer-1-Planarity

  • Seok-Hee Hong
  • Peter Eades
  • Naoki Katoh
  • Giuseppe Liotta
  • Pascal Schweitzer
  • Yusuke Suzuki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)


A graph is 1-planar if it can be embedded in the plane with at most one crossing per edge. A graph is outer-1-planar if it has an embedding in which every vertex is on the outer face and each edge has at most one crossing. We present a linear time algorithm to test whether a graph is outer-1-planar. The algorithm can be used to produce an outer-1-planar embedding in linear time if it exists.


  1. 1.
    Hong, S.H., Eades, P., Katoh, N., Liotta, G., Schweitzer, P., Suzuki, Y.: A linear-time algorithm for testing outer-1-planarity. TR-IT-IVG-2013-01. Technical Report, School of IT, University of Sydney (June 2013)Google Scholar
  2. 2.
    Auer, C., Bachmaier, C., Brandenburg, F.J., Gleißner, A., Hanauer, K., Neuwirth, D., Reislhuber, J.: Recognizing outer 1-planar graphs in linear time. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 107–118. Springer, Heidelberg (2013)Google Scholar
  3. 3.
    Ringel, G.: Ein Sechsfarbenproblem auf der Kugel. Abh. Math. Sem. Univ. Hamburg 29, 107–117 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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 (1984)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17, 427–439 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fabrici, I., Madaras, T.: The structure of 1-planar graphs. Discrete Mathematics 307, 854–865 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Suzuki, Y.: Re-embeddings of maximum 1-planar graphs. SIAM J. Discrete Math. 24, 1527–1540 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. Journal of Graph Theory 72, 30–71 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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 - (extended abstract). In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 339–345. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  10. 10.
    Eggleton, R.: Rectilinear drawings of graphs. Utilitas Mathematica 29, 149–172 (1986)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Thomassen, C.: Rectilinear drawings of graphs. Journal of Graph Theory 12, 335–341 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hong, S.H., Eades, P., Liotta, G., Poon, S.H.: Fáry’s theorem for 1-planar graphs. In: [19], pp. 335–346Google Scholar
  13. 13.
    Nagamochi, H.: Straight-line drawability of embedded graphs. Technical Report 2013-005, Department of Applied Mathematics and Physics, Kyoto University, Japan (2013)Google Scholar
  14. 14.
    Battista, G.D., Tamassia, R.: On-line maintenance of triconnected components with spqr-trees. Algorithmica 15, 302–318 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gutwenger, C., Mutzel, P.: A Linear Time Implementation of SPQR-Trees. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 77–90. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Dehkordi, H.R., Eades, P.: Every outer-1-plane graph has a right angle crossing drawing. Int. J. Comput. Geometry Appl. 22, 543–558 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Frati, F.: Straight-line drawings of outerplanar graphs in o(dn log n) area. Comput. Geom. 45, 524–533 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Knauer, K.B., Micek, P., Walczak, B.: Outerplanar graph drawings with few slopes. In: [19], pp. 323–334Google Scholar
  19. 19.
    Gudmundsson, J., Mestre, J., Viglas, T. (eds.): COCOON 2012. LNCS, vol. 7434. Springer, Heidelberg (2012)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

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

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