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Moderately Beyond Clique-Width: Reduced Component Max-Leaf and Related Parameters

Authors Édouard Bonnet , Yeonsu Chang , Julien Duron , Colin Geniet , O-joung Kwon

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Structural graph theory
  • reduced parameter
  • component max-leaf

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Abstract

Reduced parameters [BKW, JCTB '26; BKRT, SODA '22] are defined via contraction sequences. Based on this framework, we introduce the reduced component max-leaf, denoted by cml^↓, where component max-leaf is the maximum number of leaves in any spanning tree of any connected component. Reduced component max-leaf is strictly sandwiched between clique-width and reduced bandwidth, it is bounded in unit interval graphs, and unbounded in planar graphs. We design polynomial-time algorithms for problems such as Maximum Independent Set, Maximum Clique, Maximum Induced d-Regular Subgraph, and Induced Disjoint Paths in graphs given with a contraction sequence witnessing low cml^↓, unifying and extending tractability results for classes of bounded clique-width and unit interval graphs.
We get the following collapses in sparse classes of bounded cml^↓: bounded maximum degree implies bounded treewidth, whereas K_{t,t}-subgraph-freeness implies strongly sublinear treewidth; we show the latter, more generally, for classes of bounded reduced cutwidth. We establish the former result by showing that graphs with bounded cml^↓ admit balanced separators dominated by a bounded number of vertices. In contrast, there are graphs G of arbitrarily large girth and treewidth Θ(|V(G)|^{1/2}) such that cml^↓(G) ⩽ 3.
We then showcase an application of the reduced parameters to establishing non-transducibility results. We prove that for most reduced parameters p^↓ (including reduced bandwidth), the family of classes of bounded p^↓ is closed under first-order transductions. We then answer a question of [BKW '26] by showing that the 3-dimensional grids have unbounded reduced bandwidth. As the class of planar graphs (or any class of bounded genus) has bounded reduced bandwidth [BKW '26], this reproves a recent result [GPP, LICS '25; HJ, LICS '25] that planar graphs do not first-order transduce the 3-dimensional grids.

Cite As Get BibTex

Édouard Bonnet, Yeonsu Chang, Julien Duron, Colin Geniet, and O-joung Kwon. Moderately Beyond Clique-Width: Reduced Component Max-Leaf and Related Parameters. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.50

Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, France
Yeonsu Chang
  • Department of Mathematics, Hanyang University, Seoul, South Korea
Julien Duron
  • Institute of Informatics, University of Warsaw, Poland
Colin Geniet
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
O-joung Kwon
  • Department of Mathematics and Research Institute for Natural Sciences, Hanyang University, Seoul, South Korea
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea

Funding

  • Chang, Yeonsu: Supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (No. RS-2023-00211670).
  • Geniet, Colin: Supported by the Institute for Basic Science (IBS-R029-C1).
  • Kwon, O-joung: Supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (No. RS-2023-00211670) and by the Institute for Basic Science (IBS-R029-C1).

References

  1. Pierre Aboulker, Isolde Adler, Eun Jung Kim, Ni Luh Dewi Sintiari, and Nicolas Trotignon. On the tree-width of even-hole-free graphs. Eur. J. Comb., 98:103394, 2021. URL: https://doi.org/10.1016/J.EJC.2021.103394.
  2. Ambroise Baril, Miguel Couceiro, and Victor Lagerkvist. Improved bounds for twin-width parameter variants with algorithmic applications to counting graph colorings. Theory Comput. Syst., 70(1):12, 2026. URL: https://doi.org/10.1007/S00224-025-10233-Y.
  3. Pierre Bergé, Édouard Bonnet, and Hugues Déprés. Deciding twin-width at most 4 is NP-complete. In Proc. ICALP 2022, volume 229 of LIPIcs, pages 18:1-18:20. Schloss Dagstuhl, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.18.
  4. Benjamin Bergougnoux, Jan Dreier, and Lars Jaffke. A logic-based algorithmic meta-theorem for mim-width. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 3282-3304. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch125.
  5. Benjamin Bergougnoux and Mamadou Moustapha Kanté. More applications of the d-neighbor equivalence: acyclicity and connectivity constraints. SIAM J. Discrete Math., 35(3):1881-1926, 2021. URL: https://doi.org/10.1137/20M1350571.
  6. Mikolaj Bojanczyk, Michal Pilipczuk, Wojciech Przybyszewski, Marek Sokolowski, and Giannos Stamoulis. Low rank MSO. CoRR, abs/2502.08476, 2025. URL: https://doi.org/10.48550/arXiv.2502.08476.
  7. Édouard Bonnet and Julien Duron. Stretch-width. In Neeldhara Misra and Magnus Wahlström, editors, 18th International Symposium on Parameterized and Exact Computation, IPEC 2023, September 6-8, 2023, Amsterdam, The Netherlands, volume 285 of LIPIcs, pages 8:1-8:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.IPEC.2023.8.
  8. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width II: small classes. Comb. Theory, 2(2), 2022. URL: https://doi.org/10.5070/C62257876.
  9. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width III: max independent set, min dominating set, and coloring. SIAM J. Comput., 53(5):1602-1640, 2024. URL: https://doi.org/10.1137/21M142188X.
  10. Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, and Stéphan Thomassé. Twin-width VI: the lens of contraction sequences. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 1036-1056. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.45.
  11. 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. URL: https://doi.org/10.1145/3486655.
  12. Édouard Bonnet, O-joung Kwon, and David R. Wood. Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond). J. Combin. Theory Ser. B, 178:27-66, 2026. URL: https://doi.org/10.1016/j.jctb.2025.11.010.
  13. Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theoret. Comput. Sci., 511:66-76, 2013. URL: https://doi.org/10.1016/j.tcs.2013.01.009.
  14. Maria Chudnovsky. Induced subgraphs and tree decompositions. In Jim Geelen, Daniel Král, and Alexander Scott, editors, Oberwolfach report 1/2022, 2022. URL: https://doi.org/10.4171/OWR/2022/1.
  15. Bruno Courcelle, János A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  16. Marc Demange and Tínaz Ekim. A note on the np-hardness of two matching problems in induced subgrids. Discret. Math. Theor. Comput. Sci., 15(2):233-242, 2013. URL: https://doi.org/10.46298/DMTCS.606.
  17. Jan Dreier, Nikolas Mählmann, Sebastian Siebertz, and Szymon Toruńczyk. Indiscernibles and flatness in monadically stable and monadically NIP classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261, pages 125:1-125:18, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.125.
  18. Zdeněk Dvořák and Sergey Norin. Strongly sublinear separators and polynomial expansion. SIAM J. Discret. Math., 30(2):1095-1101, 2016. URL: https://doi.org/10.1137/15M1017569.
  19. Zdeněk Dvořák and Sergey Norin. Treewidth of graphs with balanced separations. J. Comb. Theory B, 137:137-144, 2019. URL: https://doi.org/10.1016/J.JCTB.2018.12.007.
  20. Fedor V. Fomin, Petr A. Golovach, Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. Compound logics for modification problems. ACM Trans. Comput. Log., 26(1):2:1-2:57, 2025. URL: https://doi.org/10.1145/3696451.
  21. Haim Gaifman. On local and non-local properties. In J. Stern, editor, Proceedings of the Herbrand Symposium, volume 107 of Studies in Logic and the Foundations of Mathematics, pages 105-135. Elsevier, 1982. URL: https://doi.org/10.1016/S0049-237X(08)71879-2.
  22. Jakub Gajarský, Michal Pilipczuk, and Filip Pokrývka. 3D-grids are not transducible from planar graphs. In 40th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2025, Singapore, June 23-26, 2025, pages 831-842. IEEE, 2025. URL: https://doi.org/10.1109/LICS65433.2025.00068.
  23. Frank Gurski and Egon Wanke. The tree-width of clique-width bounded graphs without K_n, n. In Ulrik Brandes and Dorothea Wagner, editors, Graph-Theoretic Concepts in Computer Science, 26th International Workshop, WG 2000, Konstanz, Germany, June 15-17, 2000, Proceedings, volume 1928 of Lecture Notes in Computer Science, pages 196-205. Springer, 2000. URL: https://doi.org/10.1007/3-540-40064-8_19.
  24. Petr Hliněný and Jan Jedelský. Twin-width of planar graphs is at most 8, and some related bounds. SIAM J. Discret. Math., 39(4):2003-2048, 2025. URL: https://doi.org/10.1137/23M1623823.
  25. Petr Hliněný and Jan Jedelský. Transductions of graph classes admitting product structure. In 2025 40th Annual ACM/IEEE Symposium on Logic in Computer Science - LICS 2025, pages 843-855. IEEE Comput. Soc. Press, Los Alamitos, CA, [2025] (C) 2025. URL: https://doi.org/10.1109/LICS65433.2025.00069.
  26. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-Width I. Induced path problems. Discrete Appl. Math., 278:153-168, 2020. URL: https://doi.org/10.1016/j.dam.2019.06.026.
  27. Lynn Harold Loomis and Hassler Whitney. An inequality related to the isoperimetric inequality. Bulletin of the American Mathematical Society, 55:961-962, 1949. URL: https://api.semanticscholar.org/CorpusID:123450423.
  28. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity: Graphs, Structures, and Algorithms. Springer Publishing Company, Incorporated, 2012. Google Scholar
  29. Michal Pilipczuk. Problems parameterized by treewidth tractable in single exponential time: A logical approach. In Filip Murlak and Piotr Sankowski, editors, Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings, volume 6907 of Lecture Notes in Computer Science, pages 520-531. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22993-0_47.
  30. Michal Pilipczuk, Nicole Schirrmacher, Sebastian Siebertz, Szymon Torunczyk, and Alexandre Vigny. Algorithms and data structures for first-order logic with connectivity under vertex failures. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, Paris, France, July 4-8, 2022, volume 229 of LIPIcs, pages 102:1-102:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.102.
  31. Serge Plotkin, Satish Rao, and Warren D. Smith. Shallow excluded minors and improved graph decompositions. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '94, pages 462-470, USA, 1994. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=314464.314625.
  32. Cosmin Pohoata. Unavoidable induced subgraphs of large graphs. Senior thesis, Department of Mathematics, Princeton University, 2014. URL: http://arks.princeton.edu/ark:/88435/dsp012514nk67q.
  33. Neil Robertson and P.D Seymour. Graph Minors. II. Algorithmic aspects of tree-width. Journal of Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  34. Nicole Schirrmacher, Sebastian Siebertz, Giannos Stamoulis, Dimitrios M. Thilikos, and Alexandre Vigny. Model checking disjoint-paths logic on topological-minor-free graph classes. In Pawel Sobocinski, Ugo Dal Lago, and Javier Esparza, editors, Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2024, Tallinn, Estonia, July 8-11, 2024, pages 68:1-68:12. ACM, 2024. URL: https://doi.org/10.1145/3661814.3662089.
  35. Jan Arne Telle and Andrzej Proskurowski. Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math., 10(4):529-550, 1997. URL: https://doi.org/10.1137/S0895480194275825.
  36. Martin Vatshelle. New Width Parameters of Graphs. PhD thesis, Univ. Bergen, 2012. Google Scholar
  37. Florian Zickfeld. Geometric and combinatorial structures on graphs. PhD thesis, Technical University of Berlin, December 2007. URL: https://doi.org/10.14279/depositonce-1750.
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