Skip to main content
SpringerLink
Log in

Linearity of grid minors in treewidth with applications through bidimensionality

  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minorfree graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Our result has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms.

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. J. Alber, H. L. Bodlaender, H. Fernau, T. Kloks and R. Niedermeier: Fixed parameter algorithms for dominating set and related problems on planar graphs, Algorithmica 33 (2002), 461–493.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Alber, H. Fan, M. R. Fellows, H. Fernau, R. Niedermeier, F. A. Rosamond and U. Stege: A refined search tree technique for dominating set on planar graphs, Journal of Computer and System Sciences 71(4) (2005), 385–405.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Alber, H. Fernau and R. Niedermeier: Graph separators: a parameterized view; Journal of Computer and System Sciences 67 (2003), 808–832.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Alber, H. Fernau and R. Niedermeier: Parameterized complexity: exponential speed-up for planar graph problems; Journal of Algorithms 52 (2004), 26–56.

    Article  MATH  MathSciNet  Google Scholar 

  5. N. Alon, P. Seymour and R. Thomas: A separator theorem for nonplanar graphs, Journal of the American Mathematical Society 3 (1990), 801–808.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Amir: Efficient approximation for triangulation of minimum treewidth, in: Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence (UAI-2001), San Francisco, CA, 2001, Morgan Kaufmann Publishers, pp. 7–15.

  7. B. S. Baker: Approximation algorithms for NP-complete problems on planar graphs, Journal of the Association for Computing Machinery 41 (1994), 153–180.

    MATH  MathSciNet  Google Scholar 

  8. M.-S. Chang, T. Kloks and C.-M. Lee: Maximum clique transversals, in: Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2001), vol. 2204 of Lecture Notes in Computer Science, 2001, pp. 32–43.

  9. C. Chekuri, S. Khanna and F. B. Shepherd: Edge-disjoint paths in planar graphs, in: Proceedings of the 45th Symposium on Foundations of Computer Science (FOCS 2004), 2004, pp. 71–80.

  10. E. D. Demaine, F. V. Fomin, M. Hajiaghayi and D. M. Thilikos: Bidimensional parameters and local treewidth, SIAM Journal on Discrete Mathematics 18 (2004), 501–511.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. D. Demaine, F. V. Fomin, M. Hajiaghayi and D. M. Thilikos: Fixed-parameter algorithms for (k,r)-center in planar graphs and map graphs, ACM Transactions on Algorithms 1 (2005), 33–47.

    Article  MathSciNet  Google Scholar 

  12. E. D. Demaine, F. V. Fomin, M. Hajiaghayi and D. M. Thilikos: Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs, Journal of the ACM 52 (2005), 866–893.

    Article  MathSciNet  Google Scholar 

  13. E. D. Demaine and M. Hajiaghayi: Diameter and treewidth in minor-closed graph families, revisited; Algorithmica 40(3) (2004), 211–215.

    Article  MATH  MathSciNet  Google Scholar 

  14. E. D. Demaine and M. Hajiaghayi: Equivalence of local treewidth and linear local treewidth and its algorithmic applications, in: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA’04), January 2004, pp. 833–842.

  15. E. D. Demaine and M. Hajiaghayi: Bidimensionality: New connections between FPT algorithms and PTASs; in: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), Vancouver, January 2005, pp. 590–601.

  16. E. D. Demaine and M. Hajiaghayi: Graphs excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality; in: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), Vancouver, January 2005, pp. 682–689.

  17. E. D. Demaine, M. Hajiaghayi and K. Kawarabayashi: Algorithmic graph minor theory: Decomposition, approximation, and coloring; in: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, Pittsburgh, PA, October 2005, pp. 637–646.

  18. E. D. Demaine, M. Hajiaghayi, N. Nishimura, P. Ragde and D. M. Thilikos: Approximation algorithms for classes of graphs excluding single-crossing graphs as minors, Journal of Computer and System Sciences 69 (2004), pp. 166–195.

    Article  MATH  MathSciNet  Google Scholar 

  19. E. D. Demaine, M. Hajiaghayi and D. M. Thilikos: The bidimensional theory of bounded-genus graphs, SIAM Journal on Discrete Mathematics 20(2) (2006), 357–371.

    Article  MATH  MathSciNet  Google Scholar 

  20. E. D. Demaine, M. Hajiaghayi and D. M. Thilikos: A 1.5-approximation for treewidth of graphs excluding a graph with one crossing, in: Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization (Italy, APPROX 2002), Vol. 2462 of Lecture Notes in Computer Science, 2002, pp. 67–80.

  21. E. D. Demaine, M. Hajiaghayi and D. M. Thilikos: Exponential speedup of fixed-parameter algorithms for classes of graphs excluding single-crossing graphs as minors, Algorithmica 41 (2005), 245–267.

    Article  MATH  MathSciNet  Google Scholar 

  22. R. Diestel, T. R. Jensen, K. Y. Gorbunov and C. Thomassen: Highly connected sets and the excluded grid theorem, Journal of Combinatorial Theory, Series B 75 (1999), 61–73.

    Article  MATH  MathSciNet  Google Scholar 

  23. D. Eppstein: Diameter and treewidth in minor-closed graph families, Algorithmica 27 (2000), 275–291.

    Article  MATH  MathSciNet  Google Scholar 

  24. U. Feige, M. Hajiaghayi and J. R. Lee: Improved approximation algorithms for minimum-weight vertex separators, in: Proceedings of the 37th ACM Symposium on Theory of Computing (STOC 2005), Baltimore, May 2005, pp. 563–572.

  25. F. V. Fomin and D. M. Thilikos: Dominating sets in planar graphs: Branch-width and exponential speed-up; in: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003), 2003, pp. 168–177.

  26. M. Frick and M. Grohe: Deciding first-order properties of locally tree-decomposable structures, Journal of the ACM 48 (2001), 1184–1206.

    Article  MathSciNet  Google Scholar 

  27. M. Grohe: Local tree-width, excluded minors, and approximation algorithms, Combinatorica 23(4) (2003), 613–632.

    Article  MATH  MathSciNet  Google Scholar 

  28. G. Gutin, T. Kloks and C. M. Lee: Kernels in planar digraphs, Journal of Computer and System Sciences 71(2) (2005), 174–184.

    Article  MATH  MathSciNet  Google Scholar 

  29. M. Hajiaghayi and N. Nishimura: Subgraph isomorphism, log-bounded fragmentation and graphs of (locally) bounded treewidth; in: Proc. the 27th International Symposium on Mathematical Foundations of Computer Science (Poland, 2002), 2002, pp. 305–318.

  30. I. Kanj and L. Perković: Improved parameterized algorithms for planar dominating set, in: Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science (MFCS 2002), vol. 2420 of Lecture Notes in Computer Science, 2002, pp. 399–410.

  31. P. N. Klein, S. A. Plotkin and S. Rao: Excluded minors, network decomposition, and multicommodity flow; in: Proceedings of the 25th Annual ACM Symposium on Theory of Computing (STOC’93), 1993, pp. 682–690.

  32. T. Kloks, C. M. Lee and J. Liu: New algorithms for k-face cover, k-feedback vertex set, and k-disjoint set on plane and planar graphs; in: Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2002), vol. 2573 of Lecture Notes in Computer Science, 2002, pp. 282–295.

  33. R. J. Lipton and R. E. Tarjan: Applications of a planar separator theorem, SIAM Journal on Computing 9 (1980), 615–627.

    Article  MATH  MathSciNet  Google Scholar 

  34. B. A. Reed: Tree width and tangles: a new connectivity measure and some applications; in: Surveys in combinatorics, 1997 (London), vol. 241 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 1997, pp. 87–162.

    Google Scholar 

  35. N. Robertson and P. D. Seymour: Graph minors. II. Algorithmic aspects of treewidth; Journal of Algorithms 7 (1986), 309–322.

    Article  MATH  MathSciNet  Google Scholar 

  36. N. Robertson and P. D. Seymour: Graph minors. V. Excluding a planar graph; Journal of Combinatorial Theory, Series B 41 (1986), 92–114.

    Article  MATH  MathSciNet  Google Scholar 

  37. N. Robertson and P. D. Seymour: Graph minors. XVI. Excluding a non-planar graph; Journal of Combinatorial Theory, Series B 89 (2003), 43–76.

    Article  MATH  MathSciNet  Google Scholar 

  38. N. Robertson, P. D. Seymour and R. Thomas: Quickly excluding a planar graph, Journal of Combinatorial Theory, Series B 62 (1994), 323–348.

    Article  MATH  MathSciNet  Google Scholar 

  39. P. D. Seymour: Personal communication, May 2004.

  40. P. D. Seymour and R. Thomas: Call routing and the ratcatcher, Combinatorica 14(2) (1994), 217–241.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MIT, Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA, 02139, USA

    Erik D. Demaine & Mohammadtaghi Hajiaghayi

Authors
  1. Erik D. Demaine
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammadtaghi Hajiaghayi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erik D. Demaine.

Additional information

A preliminary version of this paper appeared in the ACM-SIAM Symposium on Discrete Algorithms (SODA 2005) [16].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Demaine, E.D., Hajiaghayi, M. Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica 28, 19–36 (2008). https://doi.org/10.1007/s00493-008-2140-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-008-2140-4

Mathematics Subject Classification (2000)

Search

Navigation