Document Open Access Logo

Bridging Treewidth and Clique-Width via Cograph-Modular-Treewidth

Authors Václav Blažej , Satyabrata Jana , M. S. Ramanujan , Peter Strulo

PDF

File

Thumbnail PDF
PDF
LIPIcs.IPEC.2025.18.pdf
  • Filesize: 1.03 MB
  • 18 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Treewidth
  • Clique-width
  • Cograph
  • FPT
  • W[1]-hard

Metrics

Abstract

Many classical graph problems - such as Max Cut, Chromatic Number, Edge Dominating Set, and Hamiltonian Cycle - are polynomial-time solvable on cographs, fixed-parameter tractable (FPT) when parameterized by treewidth, but W[1]-hard when parameterized by clique-width. In contrast, Graph Isomorphism is FPT parameterized by treewidth, but for clique-width it is known to be in XP; whether it is FPT or W[1]-hard is open.
This reveals a sharp tractability gap between treewidth and clique-width. In this work, we propose a new structural graph parameter, 𝒞-modular-treewidth, which lies between treewidth and clique-width. The parameter leverages modular decomposition and restricts modules to induce graphs from a fixed class 𝒞 (e.g., cographs or edgeless graphs). By exploiting true and false twins - a hallmark of cograph-like structure - our parameter allows the design of efficient algorithms for several hard problems beyond the reach of treewidth-based methods. In this work, we show that 𝒞-modular-treewidth enables efficient solutions under suitable choices of 𝒞, opening a new pathway in the parameterized complexity landscape between treewidth and clique-width. In particular we show that 
- When parameterized by cograph-modular-treewidth, Isomorphism admits an FPT algorithm, whereas Chromatic Number remains W[1]-hard. 
- When parameterized by independent-modular-treewidth, Hamiltonian Cycle and Edge Dominating Set remain W[1]-hard.

Cite As Get BibTex

Václav Blažej, Satyabrata Jana, M. S. Ramanujan, and Peter Strulo. Bridging Treewidth and Clique-Width via Cograph-Modular-Treewidth. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.18

Author Details

Václav Blažej
  • University of Warsaw, Poland
Satyabrata Jana
  • University of Warwick, Coventry, UK
M. S. Ramanujan
  • University of Warwick, Coventry, UK
Peter Strulo
  • University of Warwick, Coventry, UK

Funding

  • Blažej, Václav: Supported by project BOBR that received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948057).
  • Jana, Satyabrata: Supported by the Engineering and Physical Sciences Research Council (grant number EP/V044621/1).
  • Ramanujan, M. S.: Supported by the Engineering and Physical Sciences Research Council (grant number EP/V044621/1).
  • Strulo, Peter: Supported by the Engineering and Physical Sciences Research Council (grant number EP/V044621/1).

References

  1. Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, 1st edition, 1974. Google Scholar
  2. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. J. Algorithms, 12(2):308-340, 1991. URL: https://doi.org/10.1016/0196-6774(91)90006-K.
  3. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci., 209(1-2):1-45, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00228-4.
  4. Hans L. Bodlaender and Klaus Jansen. On the complexity of the maximum cut problem. Nord. J. Comput., 7(1):14-31, March 2000. Google Scholar
  5. Hans L. Bodlaender and Arie M.C.A. Koster. Combinatorial optimization on graphs of bounded treewidth. Comput. J., 51(3):255-269, 2008. URL: https://doi.org/10.1093/COMJNL/BXM037.
  6. Anna Bretscher, Derek Corneil, Michel Habib, and Christophe Paul. A simple linear time LexBFS cograph recognition algorithm. SIAM Journal on Discrete Mathematics, 22(4):1277-1296, 2008. URL: https://doi.org/10.1137/060664690.
  7. Derek G. Corneil, Michel Habib, Christophe Paul, and Marc Tedder. A recursive linear time modular decomposition algorithm via LexBFS, 2024. URL: https://arxiv.org/abs/0710.3901.
  8. Derek G. Corneil and Udi Rotics. On the relationship between clique-width and treewidth. SIAM J. Comput., 34(4):825-847, 2005. URL: https://doi.org/10.1137/S0097539701385351.
  9. D.G. Corneil, H. Lerchs, and L. Stewart Burlingham. Complement reducible graphs. Discrete Applied Mathematics, 3(3):163-174, 1981. URL: https://doi.org/10.1016/0166-218X(81)90013-5.
  10. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  11. Bruno Courcelle. The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues. RAIRO Theor. Informatics Appl., 26:257-286, 1992. URL: https://doi.org/10.1051/ITA/1992260302571.
  12. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL: https://doi.org/10.1007/S002249910009.
  13. Bruno Courcelle and Mohamed Mosbah. Monadic second-order evaluations on tree-decomposable graphs. Theor. Comput. Sci., 109(1&2):49-82, 1993. URL: https://doi.org/10.1016/0304-3975(93)90064-Z.
  14. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1):77-114, 2000. URL: https://doi.org/10.1016/S0166-218X(99)00184-5.
  15. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms, volume 5. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  16. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. ACM Trans. Algorithms, 18(2):17:1-17:31, 2022. URL: https://doi.org/10.1145/3506707.
  17. Michael Dom, Daniel Lokshtanov, Saket Saurabh, and Yngve Villanger. Capacitated domination and covering: A parameterized perspective. In Parameterized and Exact Computation, Third International Workshop, IWPEC, volume 5018 of Lecture Notes in Computer Science, pages 78-90. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-79723-4_9.
  18. Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Frances A. Rosamond, Saket Saurabh, Stefan Szeider, and Carsten Thomassen. On the complexity of some colorful problems parameterized by treewidth. Inf. Comput., 209(2):143-153, 2011. URL: https://doi.org/10.1016/J.IC.2010.11.026.
  19. Jacob Focke, Dániel Marx, and Pawel Rzazewski. Counting list homomorphisms from graphs of bounded treewidth: Tight complexity bounds. ACM Trans. Algorithms, 20(2):11, 2024. URL: https://doi.org/10.1145/3640814.
  20. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Intractability of clique-width parameterizations. SIAM J. Comput., 39(5):1941-1956, 2010. URL: https://doi.org/10.1137/080742270.
  21. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Almost optimal lower bounds for problems parameterized by clique-width. SIAM J. Comput., 43(5):1541-1563, 2014. URL: https://doi.org/10.1137/130910932.
  22. Martin Grohe and Pascal Schweitzer. Isomorphism testing for graphs of bounded rank width. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 1010-1029. IEEE Computer Society, 2015. URL: https://doi.org/10.1109/FOCS.2015.66.
  23. Falko Hegerfeld and Stefan Kratsch. Tight algorithms for connectivity problems parameterized by modular-treewidth. In Daniël Paulusma and Bernard Ries, editors, Graph-Theoretic Concepts in Computer Science - 49th International Workshop, WG 2023, Fribourg, Switzerland, June 28-30, 2023, Revised Selected Papers, volume 14093 of Lecture Notes in Computer Science, pages 388-402. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-43380-1_28.
  24. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 184-192, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00026.
  25. Michael Lampis. Finer tight bounds for coloring on clique-width. SIAM Journal on Discrete Mathematics, 34(3):1538-1558, 2020. URL: https://doi.org/10.1137/19M1280326.
  26. Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth. SIAM J. Comput., 46(1):161-189, 2017. URL: https://doi.org/10.1137/140999980.
  27. Stefan Mengel. Parameterized compilation lower bounds for restricted cnf-formulas. In Nadia Creignou and Daniel Le Berre, editors, Theory and Applications of Satisfiability Testing - SAT 2016 - 19th International Conference, Bordeaux, France, July 5-8, 2016, Proceedings, volume 9710 of Lecture Notes in Computer Science, pages 3-12. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-40970-2_1.
  28. Karolina Okrasa, Marta Piecyk, and Pawel Rzazewski. Full complexity classification of the list homomorphism problem for bounded-treewidth graphs. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA, volume 173 of LIPIcs, pages 74:1-74:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.ESA.2020.74.
  29. Daniël Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model counting for CNF formulas of bounded modular treewidth. Algorithmica, 76(1):168-194, 2016. URL: https://doi.org/10.1007/S00453-015-0030-X.
  30. Sigve Hortemo Sæther and Jan Arne Telle. Between treewidth and clique-width. Algorithmica, 75(1):218-253, 2016. URL: https://doi.org/10.1007/S00453-015-0033-7.
  31. Marc Tedder, Derek G. Corneil, Michel Habib, and Christophe Paul. Simpler linear-time modular decomposition via recursive factorizing permutations. In Automata, Languages and Programming, pages 634-645, Berlin, Heidelberg, 2008. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-70575-8_52.
  32. Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic programming on tree decompositions using generalised fast subset convolution. In Algorithms - ESA 2009, volume 5757 of Lecture Notes in Computer Science, pages 566-577. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-04128-0_51.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact