Drawing Planar Graphs with Reduced Height

  • Stephane Durocher
  • Debajyoti Mondal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)


A straight-line (respectively, polyline) drawing Γ of a planar graph G on a set L k of k parallel lines is a planar drawing that maps each vertex of G to a distinct point on L k and each edge of G to a straight line segment (respectively, a polygonal chain with the bends on L k ) between its endpoints. The height of Γ is k, i.e., the number of lines used in the drawing. In this paper we compute new upper bounds on the height of polyline drawings of planar graphs using planar separators. Specifically, we show that every n-vertex planar graph with maximum degree Δ, having a simple cycle separator of size λ, admits a polyline drawing with height 4n/9 + O(λΔ), where the previously best known bound was 2n/3. Since \(\lambda\in O(\sqrt{n})\), this implies the existence of a drawing of height at most 4n/9 + o(n) for any planar triangulation with \(\Delta \in o(\sqrt{n})\). For n-vertex planar 3-trees, we compute straight-line drawings with height 4n/9 + O(1), which improves the previously best known upper bound of n/2. All these results can be viewed as an initial step towards compact drawings of planar triangulations via choosing a suitable embedding of the input graph.


  1. 1.
    Biedl, T.: Height-preserving transformations of planar graph drawings. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 380–391. Springer, Heidelberg (2014)Google Scholar
  2. 2.
    Biedl, T.: A 4-approximation for the height of drawing 2-connected outer-planar graphs. In: Erlebach, T., Persiano, G. (eds.) WAOA 2012. LNCS, vol. 7846, pp. 272–285. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Biedl, T.C.: Transforming planar graph drawings while maintaining height. CoRR abs/1308.6693 (2013),
  4. 4.
    Bonichon, N., Le Saëc, B., Mosbah, M.: Wagner’s Theorem on Realizers. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 1043–1053. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Brandenburg, F.J.: Drawing planar graphs on \(\frac{8}{9}n^2\) area. Electronic Notes in Discrete Mathematics 31, 37–40 (2008)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chrobak, M., Nakano, S.: Minimum width grid drawings of plane graphs. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 104–110. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  7. 7.
    De Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Djidjev, H., Venkatesan, S.M.: Reduced constants for simple cycle graph separation. Acta Informatica 34(3), 231–243 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dujmović, V., et al.: On the parameterized complexity of layered graph drawing. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 488–499. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Frati, F., Patrignani, M.: A note on minimum-area straight-line drawings of planar graphs. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 339–344. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Hossain, M.I., Mondal, D., Rahman, M.S., Salma, S.A.: Universal line-sets for drawing planar 3-trees. Journal of Graph Algorithms and Applications 17(2), 59–79 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Jordan, C.: Sur les assemblages de lignes. Journal für die reine und angewandte Mathematik 70(2), 185–190 (1969)Google Scholar
  13. 13.
    Mondal, D., Alam, M.J., Rahman, M.S.: Minimum-layer drawings of trees - (extended abstract). In: Katoh, N., Kumar, A. (eds.) WALCOM 2011. LNCS, vol. 6552, pp. 221–232. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Mondal, D., Nishat, R.I., Rahman, M.S., Alam, M.J.: Minimum-area drawings of plane 3-trees. Journal of Graph Algorithms and Applications 15(2), 177–204 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. Journal of Graph Theory 46(1), 39–47 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of ACM-SIAM SODA, January 22-24, pp. 138–148. ACM (1990)Google Scholar
  17. 17.
    Suderman, M.: Pathwidth and layered drawing of trees. Journal of Computational Geometry & Applications 14(3), 203–225 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Suderman, M.: Pathwidth and layered drawings of trees. International Journal of Computational Geometry and Applications 14, 203–225 (2004)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Debajyoti Mondal
    • 1
  1. 1.Department of Computer ScienceUniversity of ManitobaCanada

Personalised recommendations