The Effect of Planarization on Width

  • David EppsteinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)


We study the effects of planarization (the construction of a planar diagram D from a non-planar graph G by replacing each crossing by a new vertex) on graph width parameters. We show that for treewidth, pathwidth, branchwidth, clique-width, and tree-depth there exists a family of n-vertex graphs with bounded parameter value, all of whose planarizations have parameter value \(\varOmega (n)\). However, for bandwidth, cutwidth, and carving width, every graph with bounded parameter value has a planarization of linear size whose parameter value remains bounded. The same is true for the treewidth, pathwidth, and branchwidth of graphs of bounded degree.


  1. 1.
    Leighton, F.T.: New lower bound techniques for VLSI. In: Proceedings of 22nd Symposium on Foundations of Computer Science (FOCS 1981), pp. 1–12. IEEE (1981)Google Scholar
  2. 2.
    Di Battista, G., Didimo, W., Marcandalli, A.: Planarization of clustered graphs. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 60–74. Springer, Heidelberg (2002). CrossRefGoogle Scholar
  3. 3.
    Buchheim, C., Chimani, M., Gutwenger, C., Jünger, M., Mutzel, P.: Crossings and planarization. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization. Discrete Mathematics and its Applications, pp. 43–86. CRC Press (2014)Google Scholar
  4. 4.
    Garfunkel, S., Shank, H.: On the undecidability of finite planar graphs. J. Symb. Log. 36, 121–126 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Thulasiraman, K., Jayakumar, R., Swamy, M.N.S.: On maximal planarization of nonplanar graphs. IEEE Trans. Circ. Syst. 33(8), 843–844 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cimikowski, R.: An analysis of heuristics for graph planarization. J. Inform. Optim. Sci. 18(1), 49–73 (1997)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chuzhoy, J., Makarychev, Y., Sidiropoulos, A.: On graph crossing number and edge planarization. In: Randall, D. (ed.) Proceedings of 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA 2011). Society for Industrial and Applied Mathematics, pp. 1050–1069 (2011)Google Scholar
  8. 8.
    Borradaile, G., Eppstein, D., Zhu, P.: Planar induced subgraphs of sparse graphs. J. Graph Algorithms Appl. 19(1), 281–297 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jansen, B.M.P., Wulms, J.J.H.M.: Lower bounds for protrusion replacement by counting equivalence classes. In: Guo, J., Hermelin, D. (eds.) Proceedings of 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl-Leibniz-Zentrum für Informatik, vol. 63, pp. 17:1–17:12 (2016)Google Scholar
  10. 10.
    Nešetřil, J., Ossona de Mendez, P.: Sparsity. Graphs, Structures, and Algorithms. AC, vol. 28. Springer, Heidelberg (2012). zbMATHGoogle Scholar
  11. 11.
    Kinnersley, N.G.: The vertex separation number of a graph equals its path-width. Inform. Process. Lett. 42(6), 345–350 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chung, F.R.K., Seymour, P.D.: Graphs with small bandwidth and cutwidth. Discrete Math. 75(1–3), 113–119 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Corneil, D.G., Rotics, U.: On the relationship between clique-width and treewidth. SIAM J. Comput. 34(4), 825–847 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zarankiewicz, K.: On a problem of P. Turan concerning graphs. Fund. Math. 41, 137–145 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pach, J., Tóth, G.: Which crossing number is it, anyway? In: Motwani, R. (ed.) Proceedings of 39th Symposium on Foundations of Computer Science (FOCS 1998), pp. 617–626. IEEE (1998)Google Scholar
  17. 17.
    Kleitman, D.J.: The crossing number of \(K_{5, n}\). J. Comb. Theory 9, 315–323 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gurski, F., Wanke, E.: The tree-width of clique-width bounded graphs without \(K{_{n,n}}\). In: Brandes, U., Wagner, D. (eds.) WG 2000. LNCS, vol. 1928, pp. 196–205. Springer, Heidelberg (2000). CrossRefGoogle Scholar
  19. 19.
    Eppstein, D.: The effect of planarization on width. Electronic preprint arxiv:1708.05155 (2017)
  20. 20.
    van Geffen, B.A.M., Jansen, B.M.P., de Kroon, N.A.W.M., Morel, R., Nederlof, J.: Optimal algorithms on graphs of bounded width (and degree): cutwidth sometimes beats treewidth, but planarity does not help. Manuscript (2017)Google Scholar
  21. 21.
    Edelsbrunner, H., Guibas, L.J.: Topologically sweeping an arrangement. J. Comput. Syst. Sci. 38(1), 165–194 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nestoridis, N.V., Thilikos, D.M.: Square roots of minor closed graph classes. Discrete Appl. Math. 168, 34–39 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CaliforniaIrvineUSA

Personalised recommendations