Skip to main content
SpringerLink
Log in

Contracting Graphs to Paths and Trees

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Vertex deletion and edge deletion problems play a central role in parameterized complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain a tree or a path, respectively, by a sequence of at most k edge contractions in G. For Tree Contraction, we present a randomized 4k  n O(1) time polynomial-space algorithm, as well as a deterministic 4.98k  n O(1) time algorithm, based on a variant of the color coding technique of Alon, Yuster and Zwick. We also present a deterministic 2k+o(k)+n O(1) time algorithm for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k+3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k 2 vertices.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arora, S., Barak, B.: Computational Complexity: A Modern Approach, 1st edn. Cambridge University Press, New York (2009)

    Book  Google Scholar 

  3. Asano, T., Hirata, T.: Edge-contraction problems. J. Comput. Syst. Sci. 26(2), 197–208 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Binkele-Raible, D., Fernau, H.: Enumerate and measure: improving parameter budget management. In: Raman, V., Saurabh, S. (eds.) 5th International Symposium on Parameterized and Exact Computation, IPEC 2010. Lecture Notes in Computer Science, vol. 6478, pp. 38–49. Springer, Berlin (2010)

    Google Scholar 

  5. Bodlaender, H.L.: On disjoint cycles. Int. J. Found. Comput. Sci. 5(1), 59–68 (1994)

    Article  MATH  Google Scholar 

  6. Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) kernelization. In: 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, pp. 629–638. IEEE Computer Society, Washington (2009)

    Chapter  Google Scholar 

  8. Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011)

    Article  MATH  Google Scholar 

  9. Brouwer, A.E., Veldman, H.J.: Contractibility and NP-completeness. J. Graph Theory 11(1), 71–79 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)

    Article  MATH  Google Scholar 

  11. Cao, Y., Chen, J., Liu, Y.: On feedback vertex set new measure and new structures. In: Kaplan, H. (ed.) 12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010. Lecture Notes in Computer Science, vol. 6139, pp. 93–104. Springer, Berlin (2010)

    Chapter  Google Scholar 

  12. Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for feedback vertex set problems. J. Comput. Syst. Sci. 74(7), 1188–1198 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen, J., Kneis, J., Lu, S., Molle, D., Richter, S., Rossmanith, P., Sze, S.H., Zhang, F.: Randomized divide-and-conquer: improved path, matching, and packing algorithms. SIAM J. Comput. 38(6), 2526–2547 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cygan, M.: Deterministic parameterized connected vertex cover. In: Fomin, F. (ed.) 13th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2012. Lecture Notes in Computer Science. Springer (to appear). arXiv:1202.6642, February 2012

  16. Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: Ostrovsky, R. (ed.) 52nd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2011, pp. 150–159. IEEE, New York (2011)

    Chapter  Google Scholar 

  17. Dehne, F.K.H.A., Fellows, M.R., Langston, M.A., Rosamond, F.A., Stevens, K.: An O(2O(k) n 3) FPT algorithm for the undirected feedback vertex set problem. Theory Comput. Syst. 41(3), 479–492 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Díaz, J., Thilikos, D.M.: Fast FPT-algorithms for cleaning grids. In: Durand, B., Thomas, W. (eds.) 23rd Annual Symposium on Theoretical Aspects of Computer Science, STACS 2006. Lecture Notes in Computer Science, vol. 3884, pp. 361–371. Springer, Berlin (2006)

    Google Scholar 

  19. Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S.E., Thomas, W. (eds.) 36th International Colloquium on Automata, Languages and Programming, ICALP 2009. Lecture Notes in Computer Science, vol. 5555, pp. 378–389. Springer, Berlin (2009)

    Google Scholar 

  20. Downey, R.G., Fellows, R.: Parameterized Complexity. Monographs in Computer Science. Springer, Berlin (1999)

    Book  Google Scholar 

  21. Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. Springer, Berlin (2006)

    Google Scholar 

  22. Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  24. Golovach, P.A., Kaminski, M., Paulusma, D., Thilikos, D.M.: Increasing the minimum degree of a graph by contractions. In: Marx, D., Rossmanith, P. (eds.) 6th International Symposium on Parameterized and Exact Computation, IPEC 2011. Lecture Notes in Computer Science, vol. 7112, pp. 67–79. Springer, Berlin (2011)

    Google Scholar 

  25. Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., Wernicke, S.: Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Comput. Syst. Sci. 72(8), 1386–1396 (2006)

    Article  MATH  Google Scholar 

  26. Heggernes, P., van ’t Hof, P., Jansen, B.M.P., Kratsch, S., Villanger, Y.: Parameterized complexity of vertex deletion into perfect graph classes. In: Owe, O., Steffen, M., Telle, J.A. (eds.) 18th International Symposium on Fundamentals of Computation Theory, FCT 2011. Lecture Notes in Computer Science, vol. 6914, pp. 240–251. Springer, Berlin (2011)

    Google Scholar 

  27. Heggernes, P., van ’t Hof, P., Lévêque, B., Lokshtanov, D., Paul, C.: Contracting graphs to paths and trees. In: Marx, D., Rossmanith, P. (eds.) 6th International Symposium on Parameterized and Exact Computation, IPEC 2011. Lecture Notes in Computer Science, vol. 7112, pp. 55–66. Springer, Berlin (2011)

    Google Scholar 

  28. Heggernes, P., van ’t Hof, P., Lévêque, B., Paul, C.: Contracting chordal graphs and bipartite graphs to paths and trees. Electron. Notes Discrete Math. 37, 87–92 (2011)

    Article  Google Scholar 

  29. Heggernes, P., van ’t Hof, P., Lokshtanov, D., Paul, C.: Obtaining a bipartite graph by contracting few edges. In: Chakraborty, S., Kumar, A. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2011. LIPIcs, vol. 13, pp. 217–228. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Saarbrücken (2011)

    Google Scholar 

  30. Kawarabayashi, K., Reed, B.A.: An (almost) linear time algorithm for odd cycles transversal. In: Charikar, M. (ed.) 21st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 365–378 (2010)

    Google Scholar 

  31. Martin, B., Paulusma, D.: The computational complexity of disconnected cut and 2k 2-partition. In: Lee, J.H.M. (ed.) 17th International Conference on Principles and Practice of Constraint Programming, CP 2011. Lecture Notes in Computer Science, vol. 6876, pp. 561–575. Springer, Berlin (2011)

    Chapter  Google Scholar 

  32. Marx, D.: Chordal deletion is fixed-parameter tractable. Algorithmica 57(4), 747–768 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  33. Marx, D., Schlotter, I.: Obtaining a planar graph by vertex deletion. Algorithmica 62(3–4), 807–822 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  34. Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: 36th Annual Symposium on Foundations of Computer Science, FOCS 1995, pp. 182–191. IEEE Computer Society, Washington (1995)

    Google Scholar 

  35. Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discrete Appl. Math. 113(1), 109–128 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  36. Philip, G., Raman, V., Villanger, Y.: A quartic kernel for pathwidth-one vertex deletion. In: Thilikos, D.M. (ed.) 36th International Workshop on Graph Theoretic Concepts in Computer Science, WG 2010. Lecture Notes in Computer Science, vol. 6410, pp. 196–207 (2010)

    Google Scholar 

  37. Pilipczuk, M.: Problems parameterized by treewidth tractable in single exponential time: A logical approach. In: Murlak, F., Sankowski, P. (eds.) 36th International Symposium on Mathematical Foundations of Computer Science 2011, MFCS 2011. Lecture Notes in Computer Science, vol. 6907, pp. 520–531. Springer, Berlin (2011)

    Google Scholar 

  38. Thomassé, S.: A 4k 2 kernel for feedback vertex set. ACM Trans. Algorithms 6(2) (2010)

  39. van Bevern, R., Komusiewicz, C., Moser, H., Niedermeier, R.: Measuring indifference: unit interval vertex deletion. In: Thilikos, D.M. (ed.) 36th International Workshop on Graph Theoretic Concepts in Computer Science, WG 2010. Lecture Notes in Computer Science, vol. 6410, pp. 232–243. Springer, Berlin (2010)

    Google Scholar 

  40. van ’t Hof, P., Villanger, Y.: Proper interval vertex deletion. Algorithmica (to appear). http://dx.doi.org/10.1007/s00453-012-9661-3

  41. Watanabe, T., Ae, T., Nakamura, A.: On the removal of forbidden graphs by edge-deletion or by edge-contraction. Discrete Appl. Math. 3(2), 151–153 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  42. Watanabe, T., Ae, T., Nakamura, A.: On the NP-hardness of edge-deletion and -contraction problems. Discrete Appl. Math. 6(1), 63–78 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  43. Yannakakis, M.: Node- and edge-deletion NP-complete problems. In: Lipton, R.J., Burkhard, W.A., Savitch, W.J., Friedman, E.P., Aho, A.V. (eds.) 10th Annual ACM Symposium on Theory of Computing, STOC 1978, pp. 253–264. ACM, New York (1978)

    Google Scholar 

  44. Yannakakis, M.: The effect of a connectivity requirement on the complexity of maximum subgraph problems. J. ACM 26(4), 618–630 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  45. Yannakakis, M.: Edge-deletion problems. SIAM J. Comput. 10(2), 297–309 (1981)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Jesper Nederlof, Saket Saurabh, Erik Jan van Leeuwen and Martin Vatshelle for valuable suggestions and comments. We are also indebted to the three anonymous referees, whose detailed comments and suggestions helped us to correct small mistakes, simplify proofs and significantly improve the overall presentation of the paper.

Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, 5020, Bergen, Norway

    Pinar Heggernes & Pim van ’t Hof

  2. CNRS, LIRMM, Université Montpellier 2, Montpellier, France

    Benjamin Lévêque & Christophe Paul

  3. Department of Computer Science and Engineering, University of California San Diego, San Diego, USA

    Daniel Lokshtanov

Authors
  1. Pinar Heggernes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pim van ’t Hof
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Benjamin Lévêque
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Daniel Lokshtanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Christophe Paul
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pim van ’t Hof.

Additional information

An extended abstract of this paper has been presented at the 6th International Symposium on Parameterized and Exact Computation (IPEC 2011) [27]. This work has been supported by the Research Council of Norway (project SCOPE, 197548/V30), the French ANR project AGAPE (ANR-09-BLAN-0159) and the Languedoc-Roussillon “Chercheur d’avenir” project KERNEL.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heggernes, P., van ’t Hof, P., Lévêque, B. et al. Contracting Graphs to Paths and Trees. Algorithmica 68, 109–132 (2014). https://doi.org/10.1007/s00453-012-9670-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-012-9670-2

Keywords

Search

Navigation