Document Open Access Logo

Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs

Authors Amir Abboud , Ron Safier , Nathan Wallheimer

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2026.64.pdf
  • Filesize: 0.85 MB
  • 22 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Fine-grained complexity
  • Triangle Detection
  • H-free graphs

Metrics

Abstract

A recent paper by the authors (ITCS'26) initiates the study of the Triangle Detection problem in graphs avoiding a fixed pattern H as a subgraph and proposes a dichotomy hypothesis characterizing which patterns H make the Triangle Detection problem easier in H-free graphs than in general graphs. In this work, we demonstrate that this hypothesis is, in fact, equivalent to analogous hypotheses in two broader settings that a priori seem significantly more challenging: induced H-free graphs and colored H-free graphs. 
Our main contribution is a reduction from the induced H-free case to the non-induced H^{+}-free case, where H^{+} preserves the structural properties of H that are relevant for the dichotomy, namely 3-colorability and triangle count. A similar reduction is given for the colored case.
A key technical ingredient is a self-reduction to Unique Triangle Detection that preserves the induced H-freeness property, via a new color-coding-like reduction.

Cite As Get BibTex

Amir Abboud, Ron Safier, and Nathan Wallheimer. Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 64:1-64:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.64

Author Details

Amir Abboud
  • Weizmann Institute of Science, Rehovot, Israel
Ron Safier
  • Weizmann Institute of Science, Rehovot, Israel
Nathan Wallheimer
  • Weizmann Institute of Science, Rehovot, Israel

Funding

This work is part of the project CONJEXITY that has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation programme (grant agreement No. 101078482). Supported by a research grant from the Center for New Scientists at the Weizmann Institute of Science.
  • Abboud, Amir: Part of this work was done while visiting the Courant Institute of Mathematical Sciences at New York University.

References

  1. Amir Abboud, Shyan Akmal, and Nick Fischer. A truly subcubic combinatorial algorithm for induced 4-cycle detection. In Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3562-3599. SIAM, 2026. URL: https://doi.org/10.1137/1.9781611978971.130.
  2. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. If the current clique algorithms are optimal, so is valiant’s parser. SIAM J. Comput., 47(6):2527-2555, 2018. URL: https://doi.org/10.1137/16M1061771.
  3. Amir Abboud, Karl Bringmann, Seri Khoury, and Or Zamir. Hardness of approximation in p via short cycle removal: cycle detection, distance oracles, and beyond. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1487-1500, 2022. URL: https://doi.org/10.1145/3519935.3520066.
  4. Amir Abboud, Nick Fischer, Zander Kelley, Shachar Lovett, and Raghu Meka. New graph decompositions and combinatorial boolean matrix multiplication algorithms. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 935-943, 2024. URL: https://doi.org/10.1145/3618260.3649696.
  5. Amir Abboud, Nick Fischer, and Yarin Shechter. Faster combinatorial k-clique algorithms. In José A. Soto and Andreas Wiese, editors, LATIN 2024: Theoretical Informatics - 16th Latin American Symposium, Puerto Varas, Chile, March 18-22, 2024, Proceedings, Part I, Lecture Notes in Computer Science, pages 193-206. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-55598-5_13.
  6. Amir Abboud, Ron Safier, and Nathan Wallheimer. Triangle Detection in H-Free Graphs. In Shubhangi Saraf, editor, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026), volume 362 of Leibniz International Proceedings in Informatics (LIPIcs), pages 1:1-1:19, Dagstuhl, Germany, 2026. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Full version: https://arxiv.org/abs/2511.17224. URL: https://doi.org/10.4230/LIPIcs.ITCS.2026.1.
  7. Amir Abboud and Nathan Wallheimer. Worst-Case to Expander-Case Reductions. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), volume 251 of Leibniz International Proceedings in Informatics (LIPIcs), pages 1:1-1:23, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.1.
  8. Amir Abboud and Nathan Wallheimer. Worst-Case to Expander-Case Reductions: Derandomized and Generalized. 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 4:1-4:18, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2024.4.
  9. Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In International Colloquium on Automata, Languages, and Programming, pages 39-51. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_4.
  10. Tara Abrishami, Maria Chudnovsky, Cemil Dibek, and Paweł Rzążewski. Polynomial-time algorithm for maximum independent set in bounded-degree graphs with no long induced claws. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1448-1470. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.61.
  11. Vladimir Evgen'evich Alekseev and DV Korobitsyn. Complexity of some problems on hereditary classes of graphs. Diskretnaya Matematika, 4(4):34-40, 1992. Google Scholar
  12. Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. More asymmetry yields faster matrix multiplication. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2005-2039. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.63.
  13. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM (JACM), 42(4):844-856, 1995. URL: https://doi.org/10.1145/210332.210337.
  14. Noga Alon, Raphael Yuster, and Uri Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209-223, 1997. URL: https://doi.org/10.1007/BF02523189.
  15. Vladimir L Arlazarov, EA Dinic, MA Kronrod, and IA Faradzev. On economical construction of the transitive closure of a directed graph. In Dokl. Akad. Nauk SSSR, volume 194, pages 1209-1210, 1970. Google Scholar
  16. Arturs Backurs, Nishanth Dikkala, and Christos Tzamos. Tight hardness results for maximum weight rectangles. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, Rome, Italy, July 11-15, 2016, volume 55 of LIPIcs, pages 81:1-81:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.81.
  17. Arturs Backurs and Christos Tzamos. Improving viterbi is hard: Better runtimes imply faster clique algorithms. In International Conference on Machine Learning, pages 311-321. PMLR, 2017. URL: http://proceedings.mlr.press/v70/backurs17a.html.
  18. Nikhil Bansal and Ryan Williams. Regularity lemmas and combinatorial algorithms. Theory Comput., 8(1):69-94, 2012. URL: https://doi.org/10.4086/toc.2012.v008a004.
  19. Béla Bollobás. Modern graph theory, volume 184. Springer Science & Business Media, 1998. URL: https://doi.org/10.1007/978-1-4612-0619-4.
  20. John A Bondy and Miklós Simonovits. Cycles of even length in graphs. Journal of Combinatorial Theory, Series B, 16(2):97-105, 1974. Google Scholar
  21. Édouard Bonnet, Nicolas Bousquet, Pierre Charbit, Stéphan Thomassé, and Rémi Watrigant. Parameterized complexity of independent set in H-free graphs. Algorithmica, 82(8):2360-2394, 2020. URL: https://doi.org/10.1007/s00453-020-00730-6.
  22. Karl Bringmann, Paweł Gawrychowski, Shay Mozes, and Oren Weimann. Tree edit distance cannot be computed in strongly subcubic time (unless apsp can). ACM Transactions on Algorithms (TALG), 16(4):1-22, 2020. URL: https://doi.org/10.1145/3381878.
  23. Karl Bringmann, Allan Grønlund, and Kasper Green Larsen. A dichotomy for regular expression membership testing. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 307-318. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.36.
  24. Karl Bringmann and Philip Wellnitz. Clique-based lower bounds for parsing tree-adjoining grammars. In Juha Kärkkäinen, Jakub Radoszewski, and Wojciech Rytter, editors, 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017, Warsaw, Poland, July 4-6, 2017, volume 78 of LIPIcs, pages 12:1-12:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.CPM.2017.12.
  25. Timothy M. Chan. Speeding up the four russians algorithm by about one more logarithmic factor. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 212-217, USA, 2015. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.1137/1.9781611973730.16.
  26. Yi-Jun Chang. Hardness of rna folding problem with four symbols. Theoretical Computer Science, 757:11-26, 2019. URL: https://doi.org/10.1016/j.tcs.2018.07.010.
  27. Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on computing, 14(1):210-223, 1985. URL: https://doi.org/10.1137/0214017.
  28. 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. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2260-2278. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.139.
  29. Henry Cohn, Robert D. Kleinberg, Balázs Szegedy, and Christopher Umans. Group-theoretic algorithms for matrix multiplication. In 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005, Pittsburgh, PA, USA, October 23-25, 2005, Proceedings, pages 379-388. IEEE Computer Society, 2005. URL: https://doi.org/10.1109/SFCS.2005.39.
  30. Holger Dell, John Lapinskas, and Kitty Meeks. Approximately counting and sampling small witnesses using a colorful decision oracle. SIAM Journal on Computing, 51(4):849-899, 2022. URL: https://doi.org/10.1137/19m130604x.
  31. Guillaume Ducoffe, Michel Habib, and Laurent Viennot. Diameter, eccentricities and distance oracle computations on h-minor free graphs and graphs of bounded (distance) vapnik-chervonenkis dimension. SIAM Journal on Computing, 51(5):1506-1534, 2022. URL: https://doi.org/10.1137/20m136551x.
  32. Adrian Dumitrescu. Finding triangles or independent sets; and other dual pair approximations. Studia Scientiarum Mathematicarum Hungarica, 62(1):47-62, 2025. URL: https://doi.org/10.48550/arXiv.2105.01265.
  33. Lech Duraj, Filip Konieczny, and Krzysztof Potępa. Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs. 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 51:1-51:18, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2024.51.
  34. Tala Eagling-Vose, David C. Kutner, Felicia Lucke, Dániel Marx, Barnaby Martin, Daniël Paulusma, and Erik Jan van Leeuwen. Steiner forest for h-subgraph-free graphs, 2026. URL: https://doi.org/10.48550/arXiv.2602.21859.
  35. Paul Erdős and Tibor Gallai. On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hungar, 10:337-356, 1959. Google Scholar
  36. Peter Gartland, Daniel Lokshtanov, Tomáš Masařík, Marcin Pilipczuk, Michał Pilipczuk, and Paweł Rzążewski. Maximum weight independent set in graphs with no long claws in quasi-polynomial time. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 683-691, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3618260.3649791.
  37. Oded Goldreich. On coarse and fine approximate counting of t-cliques. In Computational Complexity and Local Algorithms: On the Interplay Between Randomness and Computation, pages 52-79. Springer, 2025. URL: https://doi.org/10.1007/978-3-031-88946-2_4.
  38. Petr A Golovach, Daniël Paulusma, and Bernard Ries. Coloring graphs characterized by a forbidden subgraph. In International Symposium on Mathematical Foundations of Computer Science, pages 443-454. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32589-2_40.
  39. Martin Grohe. Fixed-point definability and polynomial time on graphs with excluded minors. Journal of the ACM (JACM), 59(5):1-64, 2012. URL: https://doi.org/10.1145/2371656.2371662.
  40. Edin Husić, Stéphan Thomassé, and Nicolas Trotignon. The Independent Set Problem Is FPT for Even-Hole-Free Graphs. In Bart M. P. Jansen and Jan Arne Telle, editors, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019), volume 148 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1-21:12, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.IPEC.2019.21.
  41. Alon Itai and Michael Rodeh. Finding a minimum circuit in a graph. In Proceedings of the Ninth Annual ACM Symposium on Theory of Computing, STOC '77, pages 1-10, New York, NY, USA, 1977. Association for Computing Machinery. URL: https://doi.org/10.1145/800105.803390.
  42. Matthew Johnson, Barnaby Martin, Jelle J. Oostveen, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity framework for forbidden subgraphs i: The framework. Algorithmica, 87(3):429-464, January 2025. URL: https://doi.org/10.1007/s00453-024-01289-2.
  43. Marcin Kamiński. Max-cut and containment relations in graphs. Theoretical Computer Science, 438:89-95, 2012. URL: https://doi.org/10.1016/j.tcs.2012.02.036.
  44. Adam Karczmarz and Da Wei Zheng. Subquadratic algorithms in minor-free digraphs:(weighted) distance oracles, decremental reachability, and more. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4338-4351. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.147.
  45. Dmitry V Korobitsin. On the complexity of domination number determination in monogenic classes of graphs. Diskretnaya Matematika, 1992. Google Scholar
  46. Alexandr V Kostochka. The minimum hadwiger number for graphs with a given mean degree of vertices. Metody Diskret. Analiz., 38:37-58, 1982. Google Scholar
  47. Tamás Kovári, Vera Sós, and Pál Turán. On a problem of k. zarankiewicz. In Colloquium Mathematicum, volume 3, pages 50-57. Polska Akademia Nauk. Instytut Matematyczny PAN, 1954. Google Scholar
  48. Daniel Král’, Jan Kratochvíl, Zsolt Tuza, and Gerhard J Woeginger. Complexity of coloring graphs without forbidden induced subgraphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 254-262. Springer, 2001. URL: https://doi.org/10.1007/3-540-45477-2_23.
  49. Hung Le and Christian Wulff-Nilsen. Vc set systems in minor-free (di) graphs and applications. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 5332-5360. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.192.
  50. Andrea Lincoln, Virginia Vassilevska Williams, and Ryan Williams. Tight hardness for shortest cycles and paths in sparse graphs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1236-1252. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.80.
  51. Konrad Majewski, Tomáš Masařík, Jana Masaříková, Karolina Okrasa, Marcin Pilipczuk, Paweł Rzążewski, and Marek Sokołowski. Max weight independent set in graphs with no long claws: An analog of the gyárfás’ path argument. ACM Transactions on Computation Theory, 16(2):1-18, 2024. URL: https://doi.org/10.1145/3636422.
  52. Jelle J. Oostveen, Daniël Paulusma, and Erik Jan van Leeuwen. The complexity of diameter on H-free graphs. SIAM Journal on Discrete Mathematics, 39(2):1213-1245, 2025. URL: https://doi.org/10.1137/24M1677885.
  53. Ilia Nikolaevich Ponomarenko. Isomorphism problem for classes of graphs closed under contractions. Zapiski Nauchnykh Seminarov POMI, 174:147-177, 1988. Google Scholar
  54. Kevin Pratt, Yahel Uffenheimer, and Omri Weinstein. (approximate) matrix multiplication via convolutions. CoRR, abs/2510.22193, 2025. URL: https://doi.org/10.48550/arXiv.2510.22193.
  55. Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 515-524, 2013. URL: https://doi.org/10.1145/2488608.2488673.
  56. Thatchaphol Saranurak and Di Wang. Expander decomposition and pruning: Faster, stronger, and simpler. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2616-2635. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.162.
  57. Andrew Thomason. An extremal function for contractions of graphs. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 95, pages 261-265. Cambridge University Press, 1984. Google Scholar
  58. L G Valiant and V V Vazirani. NP is as easy as detecting unique solutions. In Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing, STOC '85, pages 458-463, New York, NY, USA, 1985. Association for Computing Machinery. URL: https://doi.org/10.1145/22145.22196.
  59. Virginia Vassilevska Williams and Ryan Williams. Subcubic equivalences between path, matrix and triangle problems. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 645-654. IEEE, 2010. URL: https://doi.org/10.1109/FOCS.2010.67.
  60. Huacheng Yu. An improved combinatorial algorithm for boolean matrix multiplication. Information and Computation, 261:240-247, 2018. URL: https://doi.org/10.1016/j.ic.2018.02.006.
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
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact