Skip to main content
SpringerLink
Log in

Approximate min–max relations for odd cycles in planar graphs

  • FULL LENGTH PAPER
  • Published:
Mathematical Programming Submit manuscript

Abstract

We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertex-disjoint odd cycles in a planar graph. We show that this ratio is at most 10. For the corresponding edge version of this problem, Král and Voss recently proved that this ratio is at most 2; we also give a short proof of their result.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Appel K. and Haken W. (1976). A proof of the four color theorem. Discrete Math. 16: 179–180

    Article  MATH  MathSciNet  Google Scholar 

  2. Caprara A. and Rizzi R. (2002). Packing triangles in bounded degree graphs. Inform. Process Lett. 84: 175–180

    Article  MATH  MathSciNet  Google Scholar 

  3. Dilworth R.P. (1950). A decomposition theorem for partially ordered sets. Ann. Math. 51: 161–166

    Article  MathSciNet  Google Scholar 

  4. Erdös P. and Pósa L. (1965). On independent circuits contained in a graph. Canad. J. Math. 17: 347–352

    MATH  MathSciNet  Google Scholar 

  5. Fiorini, S., Hardy, N., Reed, B., Vetta, A.: Planar graph bipartization in linear time. Discr. Appl. Math. (in press)

  6. Frank, A.: A survey on T-joins, T-cuts, and conservative weightings. In: Combinatorics, Paul Erdös is eighty, vol. 2, pp. 213–252. Keszthely, 1993, János Bolyai Mathematical Society, Budapest (1996)

  7. Garey M.R. and Johnson D.S. (1977). The rectilinear Steiner tree problem is NP-complete. SIAM. J. Appl. Math. 32: 826–834

    Article  MATH  MathSciNet  Google Scholar 

  8. Goemans M.X. and Williamson D.P. (1998). Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica 18: 37–59

    Article  MATH  MathSciNet  Google Scholar 

  9. Hadlock F. (1975). Finding a maximum cut of a planar graph in polynomial time. SIAM. J. Comput. 4: 221–225

    Article  MATH  MathSciNet  Google Scholar 

  10. Hardy, N.: Odd cycles in planar graphs. Master’s Thesis, McGill University, Montreal, Canada (June 2005)

  11. Král D. and Voss H. (2004). Edge-disjoint odd cycles in planar graphs. J. Combin. Theory. Ser. B 90: 107–120

    Article  MATH  MathSciNet  Google Scholar 

  12. Lovász L. (1975). 2-matchings and 2-covers of hypergraphs. Acta. Math. Acad. Sci. Hungar. 26: 433–444

    Article  MATH  MathSciNet  Google Scholar 

  13. Rautenbach D. and Reed B. (2001). The Erdős-Pósa property for odd cycles in highly connected graphs. Combinatorica 21: 267–278

    Article  MATH  MathSciNet  Google Scholar 

  14. Reed, B.: Mangoes and blueberries. Combinatorica 19, 267–296 (1999)

    Google Scholar 

  15. Reed B., Smith K. and Vetta A. (2004). Finding odd cycle transversals. Oper. Res. Lett. 32: 299–301

    Article  MATH  MathSciNet  Google Scholar 

  16. Robertson N., Sanders D., Seymour P. and Thomas R. (1997). The four colour theorem. J. Combin. Theory. Ser. B 70: 2–44

    Article  MATH  MathSciNet  Google Scholar 

  17. Schrijver, A.: Combinatorial Optimization: Polyhedra and efficiency, vol. C. Springer, Berlin Heidelberg New York (2003)

  18. Seymour P. (1981). On odd cuts and plane multicommodity flows. Proc. London Math Soc 42(3): 178–192

    Article  MATH  MathSciNet  Google Scholar 

  19. Vazirani V.V. (2001). Approximation algorithms. Springer, Berlin Heidelberg New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. GERAD – HEC Montréal, 3000 chemin de la Côte-Sainte-Catherine, Montréal, QC, H3T 2A7, Canada

    Samuel Fiorini

  2. Département de Mathématique, Université Libre de Bruxelles, CP 216, Bd du Triomphe, 1050, Bruxelles, Belgium

    Samuel Fiorini

  3. Department of Mathematics and Statistics, McGill University, Burnside Building, 805 Sherbrooke West, Montreal, QC, H3A 2K6, Canada

    Nadia Hardy

  4. School of Computer Science, McGill University, 3480 University, Montreal, QC, H3A 2A7, Canada

    Bruce Reed

  5. Department of Mathematics and Statistics and School of Computer Science, McGill University, Room 1118, Burnside Building, 805 Sherbrooke West, Montreal, QC, H3A 2K6, Canada

    Adrian Vetta

Authors
  1. Samuel Fiorini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nadia Hardy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bruce Reed
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Adrian Vetta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samuel Fiorini.

Additional information

This work was supported by FNRS, NSERC (PGS Master award, Canada Research Chair in Graph Theory, award 288334-04) and FQRNT (award 2005-NC-98649).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fiorini, S., Hardy, N., Reed, B. et al. Approximate min–max relations for odd cycles in planar graphs. Math. Program. 110, 71–91 (2007). https://doi.org/10.1007/s10107-006-0063-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-006-0063-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation