Advertisement

Tri-Edge-Connectivity Augmentation for Planar Straight Line Graphs

  • Marwan Al-Jubeh
  • Mashhood Ishaque
  • Kristóf Rédei
  • Diane L. Souvaine
  • Csaba D. Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5878)

Abstract

It is shown that if a planar straight line graph (Pslg) with n vertices in general position in the plane can be augmented to a 3-edge-connected Pslg, then 2n − 2 new edges are enough for the augmentation. This bound is tight: there are Pslgs with n ≥ 4 vertices such that any augmentation to a 3-edge-connected Pslg requires 2n − 2 new edges.

Keywords

Convex Hull Outer Face Simple Polygon Geometric Graph Convex Position 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abellanas, M., García, A., Hurtado, F., Tejel, J., Urrutia, J.: Augmenting the connectivity of geometric graphs. Comput. Geom. Theory Appl. 40(3), 220–230 (2008)MATHGoogle Scholar
  2. 2.
    Eswaran, K.P., Tarjan, R.E.: Augmentation problems. SIAM J. Comput. 5(4), 653–665 (1976)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Frank, A.: Augmenting graphs to meet edge-connectivity requirements. SIAM J. Discrete Math. 5(1), 22–53 (1992)CrossRefGoogle Scholar
  4. 4.
    Jackson, B., Jordán, T.: Independence free graphs and vertex connectivity augmentation. J. Comb. Theory Ser. B 94, 31–77 (2005)MATHCrossRefGoogle Scholar
  5. 5.
    Kortsarz, G., Nutov, Z.: Approximating minimum cost connectivity problems. In: Gonzalez, T.F. (ed.) Handbook of Approximation Algorithms and Metaheuristics, ch. 58. CRC Press, Boca Raton (2007)Google Scholar
  6. 6.
    Lovász, L.: Combinatorial Problems and Exercises. North-Holland, Amsterdam (1979)MATHGoogle Scholar
  7. 7.
    Mader, W.: A reduction method for edge-connectivity in graphs. Ann. Discr. Math. 3, 145–164 (1978)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Nagamochi, H., Ibaraki, T.: Augmenting edge-connectivity over the entire range in \(\tilde{O}(nm)\) time. J. Algorithms 30, 253–301 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Nagamochi, H., Ibaraki, T.: Graph connectivity and its augmentation: applications of MA orderings. Discrete Appl. Math. 123, 447–472 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Plesník, J.: Minimum block containing a given graph. Arch. Math. 27(6), 668–672 (1976)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rutter, I., Wolff, A.: Augmenting the connectivity of planar and geometric graphs. In: Proc. Conf. Topological Geom. Graph Theory, Paris, pp. 55–58 (2008)Google Scholar
  12. 12.
    Souvaine, D.L., Tóth, Cs.D.: A vertex-face assignment for plane graphs. Comput. Geom. Theory Appl. 42(5), 388–394 (2009)MATHGoogle Scholar
  13. 13.
    Tóth, Cs.D.: Connectivity augmentation in planar straight line graphs. In: Proc. Intl. Conf. on Topological and Geometric Graph Theory, Paris, pp. 51–54 (2008)Google Scholar
  14. 14.
    Tóth, Cs.D., Valtr, P.: Augmenting the edge connectivity of planar straight line graphs to three. In: Proc. 13th Spanish Meeting on Comp., Geom., Zaragoza (2009)Google Scholar
  15. 15.
    Tutte, W.T.: Connectivity in Graphs. University of Toronto Press (1966)Google Scholar
  16. 16.
    Watanabe, T., Nakamura, A.: Edge-connectivity augmentation problems. Comp. Sys. Sci. 35(1), 96–144 (1987)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Marwan Al-Jubeh
    • 1
  • Mashhood Ishaque
    • 1
  • Kristóf Rédei
    • 1
  • Diane L. Souvaine
    • 1
  • Csaba D. Tóth
    • 1
    • 2
  1. 1.Department of Computer ScienceTufts UniversityMedfordUSA
  2. 2.Department of MathematicsUniversity of CalgaryCanada

Personalised recommendations