On the Parameterized Complexity of Grid Contraction

Authors Saket Saurabh, Uéverton dos Santos Souza, Prafullkumar Tale

Thumbnail PDF


  • Filesize: 0.67 MB
  • 17 pages

Document Identifiers

Author Details

Saket Saurabh
  • The Institute Of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway
Uéverton dos Santos Souza
  • Fluminense Federal Universidade, Niterói, Brazil
Prafullkumar Tale
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany


Some part of this project was completed when the third author was a Senior Research Fellow at The Institute of Mathematical Sciences, HBNI, Chennai, India. The project started when the second and the third authors were visiting Prof. Michael Fellows at the University of Bergen, Bergen, Norway. Both the authors would like to thank Prof. Michael Fellows for the invitation.

Cite AsGet BibTex

Saket Saurabh, Uéverton dos Santos Souza, and Prafullkumar Tale. On the Parameterized Complexity of Grid Contraction. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


For a family of graphs 𝒢, the 𝒢-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists F ⊆ E(G) of size at most k such that G/F belongs to 𝒢. Here, G/F is the graph obtained from G by contracting all the edges in F. In this article, we initiate the study of Grid Contraction from the parameterized complexity point of view. We present a fixed parameter tractable algorithm, running in time c^k ⋅ |V(G)|^{{O}(1)}, for this problem. We complement this result by proving that unless ETH fails, there is no algorithm for Grid Contraction with running time c^{o(k)} ⋅ |V(G)|^{{O}(1)}. We also present a polynomial kernel for this problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Grid Contraction
  • FPT
  • Kernelization
  • Lower Bound


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Akanksha Agrawal, Fedor Fomin, Daniel Lokshtanov, Saket Saurabh, and Prafullkumar Tale. Path contraction faster than 2ⁿ. The 46th International Colloquium on Automata, Languages and Programming (ICALP 2019), 2019. Google Scholar
  2. Akanksha Agrawal, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Split contraction: The untold story. ACM Transactions on Computation Theory (TOCT), 11(3):1-22, 2019. Google Scholar
  3. Takao Asano and Tomio Hirata. Edge-Contraction Problems. Journal of Computer and System Sciences, 26(2):197-208, 1983. Google Scholar
  4. Rémy Belmonte, Petr A. Golovach, Pim Hof, and Daniël Paulusma. Parameterized complexity of three edge contraction problems with degree constraints. Acta Informatica, 51(7):473-497, 2014. Google Scholar
  5. Andries Evert Brouwer and Henk Jan Veldman. Contractibility and NP-completeness. Journal of Graph Theory, 11(1):71-79, 1987. Google Scholar
  6. Leizhen Cai and Chengwei Guo. Contracting few edges to remove forbidden induced subgraphs. In International Symposium on Parameterized and Exact Computation, pages 97-109. Springer, 2013. Google Scholar
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  8. Rod G. Downey and Michael R. Fellows. Fundamentals of Parameterized complexity. Springer-Verlag, 2013. Google Scholar
  9. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. Google Scholar
  10. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019. Google Scholar
  11. Petr A Golovach, Marcin Kamiński, Daniël Paulusma, and Dimitrios M Thilikos. Increasing the minimum degree of a graph by contractions. Theoretical computer science, 481:74-84, 2013. Google Scholar
  12. Petr A. Golovach, Pim van 't Hof, and Daniel Paulusma. Obtaining planarity by contracting few edges. Theoretical Computer Science, 476:38-46, 2013. Google Scholar
  13. Sylvain Guillemot and Dániel Marx. A faster FPT algorithm for bipartite contraction. Inf. Process. Lett., 113(22-24):906-912, 2013. Google Scholar
  14. Pinar Heggernes, Pim van 't Hof, Daniel Lokshtanov, and Christophe Paul. Obtaining a bipartite graph by contracting few edges. SIAM Journal on Discrete Mathematics, 27(4):2143-2156, 2013. Google Scholar
  15. Pinar Heggernes, Pim Van’t Hof, Benjamin Lévêque, Daniel Lokshtanov, and Christophe Paul. Contracting graphs to paths and trees. Algorithmica, 68(1):109-132, 2014. Google Scholar
  16. R. Krithika, Pranabendu Misra, Ashutosh Rai, and Prafullkumar Tale. Lossy kernels for graph contraction problems. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2016, pages 23:1-23:14, 2016. Google Scholar
  17. R Krithika, Pranabendu Misra, and Prafullkumar Tale. An FPT algorithm for contraction to cactus. In International Computing and Combinatorics Conference, pages 341-352. Springer, 2018. Google Scholar
  18. Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. On the hardness of eliminating small induced subgraphs by contracting edges. In International Symposium on Parameterized and Exact Computation, pages 243-254, 2013. Google Scholar
  19. Rolf Niedermeier. Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, 2006. Google Scholar
  20. Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura. On the removal of forbidden graphs by edge-deletion or by edge-contraction. Discrete Applied Mathematics, 3(2):151-153, 1981. Google Scholar
  21. Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura. On the NP-hardness of edge-deletion and-contraction problems. Discrete Applied Mathematics, 6(1):63-78, 1983. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail