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)

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Fabrici, I., Madaras, T.: The structure of 1-planar graphs. Discrete Mathematics 307(7-8), 854–865 (2007)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle 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)MATHCrossRefGoogle Scholar
  8. 8.
    Fáry, I.: On straight line representations of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)MATHGoogle Scholar
  9. 9.
    Hong, S., Nagamochi, H.: An algorithm for constructing star-shaped drawings of plane graphs. Comput. Geom. 43(2), 191–206 (2010)MathSciNetMATHCrossRefGoogle 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)MATHGoogle 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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Tutte, W.T.: How to draw a graph. Proc. of the London Mathematical Society 13, 743–767 (1963)MathSciNetMATHCrossRefGoogle 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

Personalised recommendations