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.

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