Document Open Access Logo

Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width

Authors Benjamin Bergougnoux , Jakub Gajarský , Grzegorz Guśpiel , Petr Hliněný , Filip Pokrývka , Marek Sokołowski

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2023.11.pdf
  • Filesize: 0.73 MB
  • 13 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Fixed parameter tractability
Keywords
  • twin-width
  • tree-width
  • excluded grid
  • sparsity

Metrics

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

Abstract

Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. Having such a sequence at hand allows to solve many otherwise hard problems efficiently. Our paper focuses on a comparison of twin-width to the more traditional tree-width on sparse graphs. Namely, we prove that if a graph G of twin-width at most 2 contains no K_{t,t} subgraph for some integer t, then the tree-width of G is bounded by a polynomial function of t. As a consequence, for any sparse graph class C we obtain a polynomial time algorithm which for any input graph G ∈ C either outputs a contraction sequence of width at most c (where c depends only on C), or correctly outputs that G has twin-width more than 2. On the other hand, we present an easy example of a graph class of twin-width 3 with unbounded tree-width, showing that our result cannot be extended to higher values of twin-width.

Cite As Get BibTex

Benjamin Bergougnoux, Jakub Gajarský, Grzegorz Guśpiel, Petr Hliněný, Filip Pokrývka, and Marek Sokołowski. Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.11

Author Details

Benjamin Bergougnoux
  • University of Warsaw, Poland
Jakub Gajarský
  • University of Warsaw, Poland
Grzegorz Guśpiel
  • Masaryk University, Brno, Czech Republic
Petr Hliněný
  • Masaryk University, Brno, Czech Republic
Filip Pokrývka
  • Masaryk University, Brno, Czech Republic
Marek Sokołowski
  • University of Warsaw, Poland

Funding

J. Gajarský and M. Sokołowski have received funding from the European Research Council (ERC) (grant agreement No 948057 - BOBR).

References

  1. Isolde Adler, Binh-Minh Bui-Xuan, Yuri Rabinovich, Gabriel Renault, Jan Arne Telle, and Martin Vatshelle. On the boolean-width of a graph: Structure and applications. In WG, volume 6410 of Lecture Notes in Computer Science, pages 159-170, 2010. Google Scholar
  2. Jungho Ahn, Kevin Hendrey, Donggyu Kim, and Sang-il Oum. Bounds for the twin-width of graphs. CoRR, abs/2110.03957, 2021. URL: https://arxiv.org/abs/2110.03957.
  3. Jakub Balabán and Petr Hliněný. Twin-width is linear in the poset width. In IPEC, volume 214 of LIPIcs, pages 6:1-6:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  4. Jakub Balabán, Petr Hliněný, and Jan Jedelský. Twin-width and transductions of proper k-mixed-thin graphs. In WG, volume 13453 of Lecture Notes in Computer Science, pages 43-55. Springer, 2022. Google Scholar
  5. Pierre Bergé, Édouard Bonnet, and Hugues Déprés. Deciding twin-width at most 4 is NP-complete. In ICALP, volume 229 of LIPIcs, pages 18:1-18:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  6. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  7. Édouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Noleen Köhler, Raul Lopes, and Stéphan Thomassé. Twin-width VIII: delineation and win-wins. In IPEC, volume 249 of LIPIcs, pages 9:1-9:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  8. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width II: small classes. In SODA, pages 1977-1996. SIAM, 2021. Google Scholar
  9. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width III: max independent set, min dominating set, and coloring. In ICALP, volume 198 of LIPIcs, pages 35:1-35:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  10. Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Pierre Simon, Stéphan Thomassé, and Szymon Torunczyk. Twin-width IV: ordered graphs and matrices. In STOC, pages 924-937. ACM, 2022. Google Scholar
  11. Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, and Stéphan Thomassé. Twin-width VI: the lens of contraction sequences. In SODA, pages 1036-1056. SIAM, 2022. Google Scholar
  12. Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, Stéphan Thomassé, and Rémi Watrigant. Twin-width and polynomial kernels. Algorithmica, 84(11):3300-3337, 2022. URL: https://doi.org/10.1007/s00453-022-00965-5.
  13. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. In FOCS, pages 601-612. IEEE, 2020. Google Scholar
  14. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. J. ACM, 69(1):3:1-3:46, 2022. Google Scholar
  15. Édouard Bonnet, Jaroslav Nesetril, Patrice Ossona de Mendez, Sebastian Siebertz, and Stéphan Thomassé. Twin-width and permutations. CoRR, abs/2102.06880, 2021. URL: https://arxiv.org/abs/2102.06880.
  16. Julia Chuzhoy and Zihan Tan. Towards tight(er) bounds for the excluded grid theorem. J. Comb. Theory, Ser. B, 146:219-265, 2021. URL: https://doi.org/10.1016/j.jctb.2020.09.010.
  17. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  18. Jan Dreier, Jakub Gajarský, Yiting Jiang, Patrice Ossona de Mendez, and Jean-Florent Raymond. Twin-width and generalized coloring numbers. Discret. Math., 345(3):112746, 2022. URL: https://doi.org/10.1016/j.disc.2021.112746.
  19. Jakub Gajarský, Michal Pilipczuk, Wojciech Przybyszewski, and Szymon Torunczyk. Twin-width and types. In ICALP, volume 229 of LIPIcs, pages 123:1-123:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  20. Jakub Gajarský, Michal Pilipczuk, and Szymon Torunczyk. Stable graphs of bounded twin-width. In Christel Baier and Dana Fisman, editors, LICS '22: 37th Annual ACM/IEEE Symposium on Logic in Computer Science, Haifa, Israel, August 2-5, 2022, pages 39:1-39:12. ACM, 2022. URL: https://doi.org/10.1145/3531130.3533356.
  21. Martin Charles Golumbic and Udi Rotics. On the clique-width of some perfect graph classes. Int. J. Found. Comput. Sci., 11(3):423-443, 2000. URL: https://doi.org/10.1142/S0129054100000260.
  22. Petr Hlinený, Filip Pokrývka, and Bodhayan Roy. FO model checking on geometric graphs. Comput. Geom., 78:1-19, 2019. URL: https://doi.org/10.1016/j.comgeo.2018.10.001.
  23. Petr Hliněný and Jan Jedelský. Twin-width of planar graphs is at most 8, and at most 6 when bipartite planar. CoRR, abs/2210.08620, 2022. Accepted to ICALP 2023. URL: https://arxiv.org/abs/2210.08620.
  24. Jaroslav Nesetril, Patrice Ossona de Mendez, Michal Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. Rankwidth meets stability. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10-13, 2021, pages 2014-2033. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.120.
  25. Michal Pilipczuk, Marek Sokolowski, and Anna Zych-Pawlewicz. Compact representation for matrices of bounded twin-width. In STACS, volume 219 of LIPIcs, pages 52:1-52:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  26. Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact