A 2-Approximation for the Height of Maximal Outerplanar Graph Drawings

  • Therese BiedlEmail author
  • Philippe Demontigny
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


In this paper, we study planar drawings of maximal outerplanar graphs with the objective of achieving small height. (We do not necessarily preserve a given planar embedding.) A recent paper gave an algorithm for such drawings that is within a factor of 4 of the optimum height. In this paper, we substantially improve the approximation factor to become 2. The main ingredient is to define a new parameter of outerplanar graphs (the umbrella depth, obtained by recursively splitting the graph into graphs called umbrellas). We argue that the height of any poly-line drawing must be at least the umbrella depth, and then devise an algorithm that achieves height at most twice the umbrella depth.


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  1. 1.
    Alam, M.J., Samee, M.A.H., Rabbi, M., Rahman, M.S.: Minimum-layer upward drawings of trees. J. Graph Algorithms Appl. 14(2), 245–267 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Batzill, J., Biedl, T.: Order-preserving drawings of trees with approximately optimal height (and small width) (2016). CoRR 1606.02233 [cs.CG] (in submission)Google Scholar
  3. 3.
    Biedl, T.: Drawing outer-planar graphs in O(n log n) area. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 54–65. Springer, Heidelberg (2002). doi: 10.1007/3-540-36151-0_6 CrossRefGoogle Scholar
  4. 4.
    Biedl, T.: Small drawings of outerplanar graphs, series-parallel graphs, and other planar graphs. Discrete and Computational Geometry 45(1), 141–160 (2011)Google Scholar
  5. 5.
    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). doi: 10.1007/978-3-642-38016-7_22 CrossRefGoogle Scholar
  6. 6.
    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). doi: 10.1007/978-3-662-45803-7_32 Google Scholar
  7. 7.
    Demaine, E.D., Huang, Y., Liao, C.-S., Sadakane, K.: Canadians Should Travel Randomly. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 380–391. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-43948-7_32 Google Scholar
  8. 8.
    Biedl, T.: Ideal tree-drawings of approximately optimal width (and small height). Journal of Graph Algorithms and Applications 21(4), 631–648 (2017)Google Scholar
  9. 9.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Demontigny, P.: A 2-approximation for the height of maximal outerplanar graphs. Master’s thesis, University of Waterloo (2016). See also CoRR report 1702.01719Google Scholar
  11. 11.
    Di Battista, G., Frati, F.: Small area drawings of outerplanar graphs. Algorithmica 54(1), 25–53 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dujmovic, V., Fellows, M., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F., Whitesides, S., Wood, D.: On the parameterized complexity of layered graph drawing. Algorithmica 52, 267–292 (2008)Google Scholar
  13. 13.
    Felsner, S., Liotta, G., Wismath, S.: Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Alg. Appl 7(4), 335–362 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Frati, F.: Straight-line drawings of outerplanar graphs in \({O}(dn\log n)\) area. Comput. Geom. 45(9), 524–533 (2012)Google Scholar
  15. 15.
    Garg, A., Rusu, A.: Area-efficient planar straight-line drawings of outerplanar graphs. Discrete Applied Mathematics 155(9), 1116–1140 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Heath, L.S., Rosenberg, A.L.: Laying out graphs using queues. SIAM Journal on Computing 21(5), 927–958 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Krug, M., Wagner, D.: Minimizing the area for planar straight-line grid drawings. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 207–212. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-77537-9_21 CrossRefGoogle Scholar
  18. 18.
    Mondal, D., Alam, M.J., Rahman, M.S.: Minimum-layer drawings of trees. In: Katoh, N., Kumar, A. (eds.) WALCOM 2011. LNCS, vol. 6552, pp. 221–232. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-19094-0_23 CrossRefGoogle Scholar
  19. 19.
    Schnyder, W.: Embedding planar graphs on the grid. In: ACM-SIAM Symposium on Discrete Algorithms (SODA 1990), pp. 138–148 (1990)Google Scholar
  20. 20.
    Suderman, M.: Pathwidth and layered drawings of trees. Intl. J. Comp. Geom. Appl, 14(3), 203–225 (2004)Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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