Fáry’s Theorem for 1-Planar Graphs

  • Seok-Hee Hong
  • Peter Eades
  • Giuseppe Liotta
  • Sheung-Hung Poon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


A plane graph is a graph embedded in a plane without edge crossings. Fáry’s theorem states that every plane graph can be drawn as a straight-line drawing, preserving the embedding of the plane graph. In this paper, we extend Fáry’s theorem to a class of non-planar graphs. More specifically, we study the problem of drawing 1-plane graphs with straight-line edges. A 1-plane graph is a graph embedded in a plane with at most one crossing per edge. We give a characterisation of those 1-plane graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. Further, we show that there are 1-plane graphs for which every straight-line drawing has exponential area. To the best of our knowledge, this is the first result to extend Fáry’s theorem to non-planar graphs.


Convex Polygon Topological Graph Linear Time Algorithm Outer Face Virtual Edge 
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.


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  1. 1.
    Borodin, O.V., Kostochka, A.V., Raspaud, A., Sopena, E.: Acyclic colouring of 1-planar graphs. Discrete Applied Mathematics 114(1-3), 29–41 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Chiba, N., Yamanouchi, T., Nishizeki, T.: Linear time algorithms for convex drawings of planar graphs. In: Progress in Graph Theory, pp. 153–173. Academic Press (1984)Google Scholar
  3. 3.
    Chrobak, M., Eppstein, D.: Planar Orientations with Low Out-degree and Compaction of Adjacency Matrices. Theor. Comput. Sci. 86(2), 243–266 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Fabrici, I., Madaras, T.: The structure of 1-planar graphs. Discrete Mathematics 307(7-8), 854–865 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall (1999)Google Scholar
  7. 7.
    Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. on Comput. 25(5), 956–997 (1996)zbMATHCrossRefGoogle Scholar
  8. 8.
    Fáry, I.: On straight line representations of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)zbMATHGoogle Scholar
  9. 9.
    Hong, S., Nagamochi, H.: An algorithm for constructing star-shaped drawings of plane graphs. Comput. Geom. 43(2), 191–206 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. on Comput. 2, 135–158 (1973)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Korzhik, V.P., Mohar, B.: Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 302–312. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Kuratowski, K.: Sur le problme des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)zbMATHGoogle Scholar
  13. 13.
    Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific (2004)Google Scholar
  14. 14.
    Pach, J., Toth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Read, R.C.: A new method for drawing a planar graph given the cyclic order of the edges at each vertex. Congr. Numer. 56, 31–44 (1987)MathSciNetGoogle Scholar
  16. 16.
    Suzuki, Y.: Optimal 1-planar graphs which triangulate other surfaces. Discrete Mathematics 310(1), 6–11 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Tutte, W.T.: How to draw a graph. Proc. of the London Mathematical Society 13, 743–767 (1963)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Seok-Hee Hong
    • 1
  • Peter Eades
    • 1
  • Giuseppe Liotta
    • 2
  • Sheung-Hung Poon
    • 3
  1. 1.University of SydneyAustralia
  2. 2.University of PerugiaItaly
  3. 3.National Tsing Hua UniversityTaiwan

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