Document Open Access Logo

Modification to Planarity is Fixed Parameter Tractable

Authors Fedor V. Fomin , Petr A. Golovach , Dimitrios M. Thilikos

PDF
Thumbnail PDF

File

LIPIcs.STACS.2019.28.pdf
  • Filesize: 0.82 MB
  • 17 pages

Document Identifiers

Author Details

Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
Dimitrios M. Thilikos
  • AlGCo project-team, LIRMM, Université de Montpellier, CNRS, Montpellier, France

Funding

  • Fomin, Fedor V.: Supported by the Research Council of Norway via the projects "CLASSIS" and "MULTIVAL". Supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs, via the Franco-Norwegian project PHC AURORA 2019.
  • Golovach, Petr A.: Supported by the Research Council of Norway via the projects "CLASSIS" and "MULTIVAL". Supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs, via the Franco-Norwegian project PHC AURORA 2019.
  • Thilikos, Dimitrios M.: Supported by projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010). Supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs, via the Franco-Norwegian project PHC AURORA 2019.

Cite As Get BibTex

Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Modification to Planarity is Fixed Parameter Tractable. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.STACS.2019.28

Abstract

A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G|^{2}) steps. We also present several applications of our approach to related problems.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Modification problems
  • Planar graphs
  • Irrelevant vertex technique

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Isolde Adler, Stavros G. Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Tight Bounds for Linkages in Planar Graphs. In Proceedings of the 38th International Colloquium of Automata, Languages and Programming (ICALP), volume 6755 of Lecture Notes in Comput. Sci., pages 110-121. Springer, 2011. Google Scholar
  2. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. Optimal Algorithms for Hitting (Topological) Minors on Graphs of Bounded Treewidth. In 12th International Symposium on Parameterized and Exact Computation, IPEC 2017, September 6-8, 2017, Vienna, Austria, pages 4:1-4:12, 2017. Google Scholar
  3. Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (Meta) Kernelization. J. ACM, 63(5):44:1-44:69, 2016. Google Scholar
  4. Dimitris Chatzidimitriou, Archontia C. Giannopoulou, Spyridon Maniatis, Clément Requilé, Dimitrios M. Thilikos, and Dimitris Zoros. FPT algorithms for plane completion problems. In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, pages 26:1-26:13, 2016. Google Scholar
  5. Bruno Courcelle. The Expression of Graph Properties and Graph Transformations in Monadic Second-Order Logic. Handbook of Graph Grammars, pages 313-400, 1997. Google Scholar
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
  7. Marek Cygan, Dániel Marx, Marcin Pilipczuk, and Michał Pilipczuk. The planar directed k-Vertex-Disjoint Paths problem is fixed-parameter tractable. In Proceedings of the 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 197-207. IEEE, 2013. Google Scholar
  8. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting Forbidden Minors: Approximation and Kernelization. SIAM Journal on Discrete Mathematics, 30(1):383-410, 2016. URL: http://dx.doi.org/10.1137/140997889.
  9. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms. In Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS), pages 470-479. IEEE, 2012. Google Scholar
  10. Archontia C Giannopoulou, Bart MP Jansen, Daniel Lokshtanov, and Saket Saurabh. Uniform kernelization complexity of hitting forbidden minors. ACM Transactions on Algorithms, 13(3):35, 2017. Google Scholar
  11. Archontia C Giannopoulou, Michał Pilipczuk, Dimitrios M Thilikos, Jean-Florent Raymond, and Marcin Wrochna. Linear kernels for edge deletion problems to immersion-closed graph classes. arXiv preprint arXiv:1609.07780, 2016. Google Scholar
  12. Petr A. Golovach, Marcin Kaminski, Spyridon Maniatis, and Dimitrios M. Thilikos. The Parameterized Complexity of Graph Cyclability. SIAM J. Discrete Math., 31(1):511-541, 2017. URL: http://dx.doi.org/10.1137/141000014.
  13. Petr A. Golovach, Pim van 't Hof, and Daniël Paulusma. Obtaining planarity by contracting few edges. Theor. Comput. Sci., 476:38-46, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2012.12.041.
  14. Martin Grohe, Ken-ichi Kawarabayashi, Dániel Marx, and Paul Wollan. Finding topological subgraphs is fixed-parameter tractable. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pages 479-488. ACM, 2011. Google Scholar
  15. Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. A Near-optimal Planarization Algorithm. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14, pages 1802-1811. SIAM, 2014. URL: http://dl.acm.org/citation.cfm?id=2634074.2634204.
  16. Ken-ichi Kawarabayashi. Planarity Allowing Few Error Vertices in Linear Time. In Proceedings of the 50th Annual Symposium on Foundations of Computer Science (FOCS), pages 639-648. IEEE, 2009. Google Scholar
  17. Ken-ichi Kawarabayashi and Yusuke Kobayashi. The induced disjoint path problem. In 13th Conference on Integer Programming and Combinatorial Optimization, IPCO 2008, volume 5035 of Lecture Notes in Computer Science, pages 47-61. Springer, Berlin, 2008. Google Scholar
  18. Ken-ichi Kawarabayashi and Bojan Mohar. Graph and map isomorphism and all polyhedral embeddings in linear time. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC), pages 471-480. ACM, 2008. Google Scholar
  19. Ken-ichi Kawarabayashi, Bojan Mohar, and Bruce A. Reed. A Simpler Linear Time Algorithm for Embedding Graphs into an Arbitrary Surface and the Genus of Graphs of Bounded Tree-Width. In Proceedings of the 49th Annual Symposium on Foundations of Computer Science (FOCS), pages 771-780. IEEE Computer Society, 2008. Google Scholar
  20. Ken-ichi Kawarabayashi and Bruce A. Reed. Odd cycle packing. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, pages 695-704, 2010. Google Scholar
  21. Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear Kernels and Single-Exponential Algorithms Via Protrusion Decompositions. ACM Transactions on Algorithms, 12(2):21:1-21:41, 2016. URL: http://dx.doi.org/10.1145/2797140.
  22. Dániel Marx. Chordal Deletion is Fixed-Parameter Tractable. Algorithmica, 57(4):747-768, 2010. URL: http://dx.doi.org/10.1007/s00453-008-9233-8.
  23. Dániel Marx and Ildikó Schlotter. Obtaining a Planar Graph by Vertex Deletion. Algorithmica, 62(3-4):807-822, 2012. Google Scholar
  24. Neil Robertson and Paul D. Seymour. Graph Minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(2):92-114, 1986. Google Scholar
  25. Neil Robertson and Paul D. Seymour. Graph Minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B, 63(1):65-110, 1995. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact