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Tree-Independence Number of P₅-Free Graphs with No Large Bicliques

Authors Václav Blažej , J. Pascal Gollin , Tomáš Hons , Tomáš Masařík , Martin Milanič , Paweł Rzążewski , Ondřej Suchý , Alexandra Wesolek

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
Keywords
  • tree-independence number
  • independence degeneracy
  • independence treewidth
  • P₅-free graphs

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Abstract

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties, but the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique K_{𝓁,𝓁} forces tree-independence number at least 𝓁. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers t and 𝓁, {P_t,K_{𝓁,𝓁}}-free graphs have bounded tree-independence number. We prove this conjecture for t = 5 by showing that every {P₅,K_{𝓁,𝓁}}-free graph has tree-independence number at most 4𝓁. We also obtain related bounds for the weaker parameter of α-degeneracy.

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Václav Blažej, J. Pascal Gollin, Tomáš Hons, Tomáš Masařík, Martin Milanič, Paweł Rzążewski, Ondřej Suchý, and Alexandra Wesolek. Tree-Independence Number of P₅-Free Graphs with No Large Bicliques. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 77:1-77:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.77

Author Details

Václav Blažej
  • Institute of Informatics, University of Warsaw, Poland
J. Pascal Gollin
  • FAMNIT, University of Primorska, Koper, Slovenia
Tomáš Hons
  • Computer Science Institute, Charles University, Czech Republic
  • Institute of Computer Science, Czech Academy of Sciences, Czech Republic
Tomáš Masařík
  • Institute of Informatics, University of Warsaw, Poland
Martin Milanič
  • FAMNIT and IAM, University of Primorska, Koper, Slovenia
Paweł Rzążewski
  • Warsaw University of Technology, Poland
Ondřej Suchý
  • Faculty of Information Technology, Czech Technical University in Prague, Czech Republic
Alexandra Wesolek
  • CNRS, LaBRI, University of Bordeaux, France

Funding

  • Blažej, Václav: Supported by the project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948057).
  • Gollin, J. Pascal: Supported in part by the Slovenian Research and Innovation Agency (research project N1-0370).
  • Hons, Tomáš: Supported by Project 24-12591M of the Czech Science Foundation (GAČR).
  • Masařík, Tomáš: Supported by the Polish National Science Centre SONATA-17 grant number 2021/43/D/ST6/03312.
  • Milanič, Martin: Supported in part by the Slovenian Research and Innovation Agency (I0-0035, research program P1-0285 and research projects J1-60012, J1-70035, J1-70046, and N1-0370).
  • Rzążewski, Paweł: Supported by the National Science Centre grant number 2024/54/E/ST6/00094.
  • Suchý, Ondřej: Co-funded by the European Union under the project Robotics and advanced industrial production (reg. no. CZ.02.01.01/00/22_008/0004590).
  • Wesolek, Alexandra: Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID:390685689).

Acknowledgements

The project was initiated at the workshop Homonolo 2024. We are grateful to the organizers and other participants, especially Peter Strulo, for productive atmosphere and inspiring discussions. We are also grateful to Olivia Milanič whose patience significantly helped in finishing this paper.

References

  1. Tara Abrishami, Marcin Briański, Jadwiga Czyżewska, Rose McCarty, Martin Milanič, Paweł Rzążewski, and Bartosz Walczak. Excluding a clique or a biclique in graphs of bounded induced matching treewidth. SIAM J. Discrete Math., 39(2):1189-1200, 2025. URL: https://doi.org/10.1137/24M1659960.
  2. Jungho Ahn, Jochen Pascal Gollin, Tony Huynh, and O-joung Kwon. A coarse Erdős-Pósa theorem. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 3363-3381. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.109.
  3. Aistis Atminas, Vadim V. Lozin, and Igor Razgon. Linear time algorithm for computing a small biclique in graphs without long induced paths. In Fedor V. Fomin and Petteri Kaski, editors, Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 2012. Proceedings, Lecture Notes in Computer Science, pages 142-152. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31155-0_13.
  4. Gábor Bacsó, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Zsolt Tuza, and Erik Jan van Leeuwen. Subexponential-time algorithms for maximum independent set in P_t-free and broom-free graphs. Algorithmica, 81(2):421-438, 2019. URL: https://doi.org/10.1007/s00453-018-0479-5.
  5. 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.
  6. Václav Blažej, J. Pascal Gollin, Tomáš Hons, Tomáš Masařík, Martin Milanič, Paweł Rzążewski, Ondřej Suchý, and Alexandra Wesolek. Tree-independence number of P₅-free graphs with no large bicliques. Preprint, 2026. URL: https://doi.org/10.48550/arXiv.2605.03965.
  7. Hans L. Bodlaender, Hirotaka Ono, and Yota Otachi. Degree-constrained orientation of maximum satisfaction: graph classes and parameterized complexity. Algorithmica, 80(7):2160-2180, 2018. URL: https://doi.org/10.1007/s00453-017-0399-9.
  8. Marthe Bonamy, Nicolas Bousquet, Michał Pilipczuk, Paweł Rzążewski, Stéphan Thomassé, and Bartosz Walczak. Degeneracy of P_t-free and C_⩾ t-free graphs with no large complete bipartite subgraphs. J. Combin. Theory Ser. B, 152:353-378, 2022. URL: https://doi.org/10.1016/j.jctb.2021.10.005.
  9. Nick Brettell, Jake Horsfield, Andrea Munaro, and Daniël Paulusma. List k-colouring P_ t-free graphs: a mim-width perspective. Inform. Process. Lett., 173:Paper No. 106168, 7, 2022. URL: https://doi.org/10.1016/j.ipl.2021.106168.
  10. Steven Chaplick, Martin Töpfer, Jan Voborník, and Peter Zeman. On H-topological intersection graphs. Algorithmica, 83(11):3281-3318, 2021. URL: https://doi.org/10.1007/s00453-021-00846-3.
  11. Mujin Choi, Claire Hilaire, Martin Milanič, and Sebastian Wiederrecht. Excluding an induced wheel minor in graphs without large induced stars. In Henning Fernau and Philipp Kindermann, editors, Graph-Theoretic Concepts in Computer Science - 51st International Workshop, WG 2025, Otzenhausen, Germany, June 11-13, 2025, Revised Selected Papers, Lecture Notes in Computer Science, pages 135-148. Springer, 2025. URL: https://doi.org/10.1007/978-3-032-11835-6_10.
  12. Maria Chudnovsky. The Erdős-Hajnal conjecture - a survey. J. Graph Theory, 75(2):178-190, 2014. URL: https://doi.org/10.1002/jgt.21730.
  13. Maria Chudnovsky, Julien Codsi, J. Pascal Gollin, Martin Milanič, and Varun Sivashankar. Tree-independence number and forbidden induced subgraphs: excluding a 6-vertex path and a (2,t)-biclique. Preprint, 2026. URL: https://doi.org/10.48550/arXiv.2604.01999.
  14. Maria Chudnovsky, Julien Codsi, Daniel Lokshtanov, Martin Milanič, and Varun Sivashankar. Tree independence number V. Walls and claws. Preprint, 2025. URL: https://doi.org/10.48550/arXiv.2501.14658.
  15. Maria Chudnovsky, J. Pascal Gollin, Matjaž Krnc, and Martin Milanič. Dominated balanced separators in wheel-induced-minor-free graphs. Preprint, 2025. URL: https://doi.org/10.48550/arXiv.2512.12329.
  16. Maria Chudnovsky, Marcin Pilipczuk, Michał Pilipczuk, and Stéphan Thomassé. Quasi-polynomial time approximation schemes for the maximum weight independent set problem in H-free graphs. SIAM J. Comput., 53(1):47-86, 2024. URL: https://doi.org/10.1137/20M1333778.
  17. Clément Dallard, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Martin Milanič. Computing tree decompositions with small independence number. ACM Trans. Algorithms, 22(1):Art. 12, 25, 2026. Google Scholar
  18. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. I. Graph classes with a forbidden structure. SIAM J. Discrete Math., 35(4):2618-2646, 2021. URL: https://doi.org/10.1137/20M1352119.
  19. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. II. Tree-independence number. J. Combin. Theory Ser. B, 164:404-442, 2024. URL: https://doi.org/10.1016/j.jctb.2023.10.006.
  20. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure. J. Combin. Theory Ser. B, 167:338-391, 2024. URL: https://doi.org/10.1016/j.jctb.2024.03.005.
  21. Clément Dallard, Matjaž Krnc, O-joung Kwon, Martin Milanič, Andrea Munaro, Kenny Štorgel, and Sebastian Wiederrecht. Treewidth versus clique number. IV. Tree-independence number of graphs excluding an induced star. Preprint, 2024. URL: https://doi.org/10.48550/arXiv.2402.11222.
  22. Milind Dawande, Pinar Keskinocak, Jayashankar M. Swaminathan, and Sridhar R. Tayur. On bipartite and multipartite clique problems. J. Algorithms, 41(2):388-403, 2001. URL: https://doi.org/10.1006/JAGM.2001.1199.
  23. Xiying Du and Rose McCarty. A survey of degree-boundedness. European J. Combin., 129:Paper No. 104092, 26, 2025. URL: https://doi.org/10.1016/j.ejc.2024.104092.
  24. Louis Esperet, Laetitia Lemoine, Frédéric Maffray, and Grégory Morel. The chromatic number of P₅,K₄-free graphs. Discrete Math., 313(6):743-754, 2013. URL: https://doi.org/10.1016/j.disc.2012.12.019.
  25. Peter Gartland. Quasi-Polynomial Time Techniques for Independent Set and Beyond in Hereditary Graph Classes. PhD thesis, University of California, Santa Barbara, USA, 2023. URL: https://www.escholarship.org/uc/item/0kk6d2jv.
  26. András Gyárfás. Problems from the world surrounding perfect graphs. Zastos. Mat., 19(3-4):413-441, 1987. Google Scholar
  27. András Gyárfás and Manouchehr Zaker. On (δ,χ)-bounded families of graphs. Electron. J. Combin., 18(1):Paper 108, 8, 2011. URL: https://doi.org/10.37236/595.
  28. Sepehr Hajebi. Polynomial bounds for pathwidth. Preprint, 2025. URL: https://doi.org/10.48550/arXiv.2510.19120.
  29. Sepehr Hajebi and Sophie Sprikl. Tree-alpha and excluding finitely many graphs. Preprint, 2026. URL: https://doi.org/10.48550/arXiv.2605.01223.
  30. Danny Hermelin, Julián Mestre, and Dror Rawitz. Optimization problems in dotted interval graphs. Discrete Appl. Math., 174:66-72, 2014. URL: https://doi.org/10.1016/j.dam.2014.04.014.
  31. Claire Hilaire, Martin Milanič, and Đorđe Vasić. Treewidth Versus Clique Number. V. Further Connections With Tree-Independence Number. Journal of Graph Theory, 112(3):337-351, 2026. URL: https://doi.org/10.1002/jgt.70036.
  32. Tommy R. Jensen and Bjarne Toft. Graph coloring problems. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., New York, 1995. Google Scholar
  33. H. A. Kierstead and S. G. Penrice. Radius two trees specify χ-bounded classes. J. Graph Theory, 18(2):119-129, 1994. URL: https://doi.org/10.1002/jgt.3190180203.
  34. Paloma T. Lima, Martin Milanič, Peter Muršič, Karolina Okrasa, Paweł Rzążewski, and Kenny Štorgel. Tree decompositions meet induced matchings: beyond Max Weight Independent Set. In Timothy Chan, Johannes Fischer, John Iacono, and Grzegorz Herman, editors, 32nd Annual European Symposium on Algorithms (ESA 2024), volume 308 of Leibniz International Proceedings in Informatics (LIPIcs), pages 85:1-85:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, 2024. URL: https://doi.org/10.4230/LIPIcs.ESA.2024.85.
  35. Daniel Lokshtanov, Michal Pilipczuk, and Paweł Rzążewski. Finding large sparse induced subgraphs in graphs of small (but not very small) tree-independence number. Preprint, 2026. URL: https://doi.org/10.48550/arXiv.2601.15861.
  36. Daniel Lokshtanov, Martin Vatshelle, and Yngve Villanger. Independent set in P₅-free graphs in polynomial time. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 570-581. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.43.
  37. Vadim Lozin. Hereditary classes of graphs: a parametric approach. Discrete Appl. Math., 325:134-151, 2023. URL: https://doi.org/10.1016/j.dam.2022.10.016.
  38. Vadim Lozin and Igor Razgon. Tree-width dichotomy. European J. Combin., 103:Paper No. 103517, 8, 2022. URL: https://doi.org/10.1016/j.ejc.2022.103517.
  39. Tung Nguyen, Alex Scott, and Paul Seymour. Induced subgraph density. VII. The five-vertex path. Proc. Lond. Math. Soc. (3), 132(3):Paper No. e70133, 2026. URL: https://doi.org/10.1112/plms.70133.
  40. Tung H. Nguyen. Polynomial χ-boundedness for excluding P₅. Preprint, 2025. URL: https://doi.org/10.48550/arXiv.2512.24907.
  41. Alex Scott and Paul Seymour. A survey of χ-boundedness. J. Graph Theory, 95(3):473-504, 2020. URL: https://doi.org/10.1002/jgt.22601.
  42. Alex Scott, Paul Seymour, and Sophie Spirkl. Polynomial bounds for chromatic number. IV: A near-polynomial bound for excluding the five-vertex path. Combinatorica, 43(5):845-852, 2023. URL: https://doi.org/10.1007/s00493-023-00015-w.
  43. D. P. Sumner. Subtrees of a graph and the chromatic number. In The theory and applications of graphs (Kalamazoo, Mich., 1980), pages 557-576. Wiley, New York, 1981. Google Scholar
  44. Mihalis Yannakakis. Node- and edge-deletion NP-complete problems. In Richard J. Lipton, Walter A. Burkhard, Walter J. Savitch, Emily P. Friedman, and Alfred V. Aho, editors, Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1-3, 1978, San Diego, California, USA, pages 253-264. ACM, 1978. URL: https://doi.org/10.1145/800133.804355.
  45. Yuli Ye and Allan Borodin. Elimination graphs. ACM Trans. Algorithms, 8(2):14:1-14:23, 2012. URL: https://doi.org/10.1145/2151171.2151177.
  46. Nikola Yolov. Minor-matching hypertree width. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 219-233. SIAM, Philadelphia, PA, 2018. URL: https://doi.org/10.1137/1.9781611975031.16.
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