Triangulating Planar Graphs While Keeping the Pathwidth Small

  • Therese BiedlEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)


Any simple planar graph can be triangulated, i.e., we can add edges to it, without adding multi-edges, such that the result is planar and all faces are triangles. In this paper, we study the problem of triangulating a planar graph without increasing the pathwidth by much. We show that if a planar graph has pathwidth k, then we can triangulate it so that the resulting graph has pathwidth O(k) (where the factors are 1, 8 and 16 for 3-connected, 2-connected and arbitrary graphs). With similar techniques, we also show that any outer-planar graph of pathwidth k can be turned into a maximal outer-planar graph of pathwidth at most \(4k+4\). The previously best known result here was \(16k+15\).


  1. 1.
    Babu, J., Basavaraju, M., Chandran, L.S., Rajendraprasad, D.: 2-connecting outer-planar graphs without blowing up the pathwidth. Theor. Comput. Sci. 554, 119–134 (2014)MathSciNetCrossRefzbMATHGoogle 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.: On triangulating \(k\)-outer-planar graphs. Discrete Appl. Math. 181, 275–279 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Biedl, T., Kant, G., Kaufmann, M.: On triangulating planar graphs under the four-connectivity constraint. Algorithmica 19(4), 427–446 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biedl, T., Velázquez, L.E.R.: Drawing planar 3-trees with given face areas. Comput. Geom. Theor. Appl. 46(3), 276–285 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L.: Treewidth: algorithmic techniques and results. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  7. 7.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25(5), 956–997 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Felsner, S., Liotta, G., Wismath, S.: Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Algorithms Appl. 7(4), 335–362 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 1st edn. Academic Press, New York (1980)zbMATHGoogle Scholar
  11. 11.
    Gutwenger, C., Mutzel, P., Zey, B.: On the hardness and approximability of planar biconnectivity augmentation. In: Ngo, H.Q. (ed.) COCOON 2009. LNCS, vol. 5609, pp. 249–257. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Kant, G., Bodlaender, H.L.: Triangulating planar graphs while minimizing the maximum degree. In: Nurmi, O., Ukkonen, E. (eds.) SWAT 1992. LNCS, vol. 621, pp. 258–271. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  13. 13.
    Schnyder, W.: Embedding planar graphs on the grid. In: ACM-SIAM Symposium on Discrete Algorithms (SODA 1990), pp. 138–148 (1990)Google Scholar
  14. 14.
    Suderman, M.: Pathwidth and layered drawings of trees. Int. J. Comput. Geom. Appl. 14(3), 203–225 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

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

Personalised recommendations