A Polynomial Bound for Untangling Geometric Planar Graphs

  • Prosenjit Bose
  • Vida Dujmović
  • Ferran Hurtado
  • Stefan Langerman
  • Pat Morin
  • David R. Wood
Article

Abstract

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002) asked if every n-vertex geometric planar graph can be untangled while keeping at least nε vertices fixed. We answer this question in the affirmative with ε=1/4. The previous best known bound was \(\Omega(\sqrt{\log n/\log\log n})\) . We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least \(\Omega(\sqrt{n})\) vertices fixed, while the best upper bound was \(\mathcal{O}((n\log n)^{2/3})\) . We answer a question of Spillner and Wolff (http://arxiv.org/abs/0709.0170) by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than \(3(\sqrt{n}-1)\) vertices fixed.

Keywords

Geometric graphs Untangling Crossings 

References

  1. 1.
    Abellanas, M., Hurtado, F., Ramos, P.: Tolerancia de arreglos de segmentos. In: Proc. VI Encuentros de Geometría Computacional, pp. 77–84 (1995) Google Scholar
  2. 2.
    Cibulka, J.: Untangling polygons and graphs. In: Topological and Geometric Graph Theory. Electronic Notes in Discrete Mathematics, vol. 31, pp. 207–211 (2008) Google Scholar
  3. 3.
    de Fraysseix, H., Pach, J., Pollac, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51(2), 161–166 (1950) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 464–470 (1935) Google Scholar
  6. 6.
    Fáry, I.: On straight line representation of planar graphs. Acta Univ. Szeged. Sect. Sci. Math. 11, 229–233 (1948) MathSciNetGoogle Scholar
  7. 7.
    Goaoc, X., Kratochvil, J., Okamoto, Y., Shin, C.-S., Wolff, A.: Moving vertices to make drawings plane. In: Proc. 15th International Symp. on Graph Drawing (GD’07). Lecture Notes in Computer Science, vol. 4875, pp. 101–112. Springer, Berlin (2008). Also in http://arxiv.org/abs/0706.1002 Google Scholar
  8. 8.
    Hong, S.-H., Nagamochi, H.: Convex drawings of graphs with non-convex boundary constraints. Discrete Appl. Math. 156(12), 2368–2380 (2008) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kang, M., Pikhurko, O., Ravsky, A., Schacht, M., Verbitsky, O.: Obfuscated drawings of planar graphs (2008). http://arxiv.org/abs/0803.0858
  10. 10.
    Kang, M., Schacht, M., Verbitsky, O.: How much work does it take to straighten a plane graph out? (2007). http://arxiv.org/abs/0707.3373
  11. 11.
    Pach, J., Tardos, G.: Untangling a polygon. Discrete Comput. Geom. 28(4), 585–592 (2002) MATHMathSciNetGoogle Scholar
  12. 12.
    Ramos, P.: Tolerancia de estructuras geométricas y combinatorias. Ph.D. thesis, Universidad Politécnica de Madrid, Madrid, Spain (1995) Google Scholar
  13. 13.
    Ravsky, A., Verbitsky, O.: On collinear sets in straight line drawings (2008). http://arxiv.org/abs/0806.0253
  14. 14.
    Spillner, A., Wolff, A.: Untangling a planar graph. In: Proc. 34th Internat. Conf. on Current Trends in Theory and Practice of Computer Science (SOFSEM’08). Lecture Notes in Computer Science, vol. 4910, pp. 473–484. Springer, Berlin (2008). Also in http://arxiv.org/abs/0709.0170 Google Scholar
  15. 15.
    Verbitsky, O.: On the obfuscation complexity of planar graphs. Theor. Comput. Sci. 396(1–3), 294–300 (2008) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Wagner, K.: Über eine Eigenschaft der ebene Komplexe. Math. Ann. 114, 570–590 (1937) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Vida Dujmović
    • 2
  • Ferran Hurtado
    • 3
  • Stefan Langerman
    • 4
  • Pat Morin
    • 1
  • David R. Wood
    • 5
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain
  4. 4.FNRS, Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  5. 5.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia

Personalised recommendations