Document Open Access Logo

Computing the (k+2)-Edge-Connected Components in k-Edge-Connected Digraphs in Subquadratic Time

Authors Loukas Georgiadis , Evangelos Kipouridis , Evangelos Kosinas , Charis Papadopoulos , Nikos Parotsidis

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2026.95.pdf
  • Filesize: 0.89 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Network flows
Keywords
  • Graph connectivity
  • edge-connected components
  • directed edge-cuts

Metrics

Abstract

Computing edge-connected components in directed and undirected graphs is a fundamental and well-studied problem in graph algorithms. In a very recent breakthrough, Korhonen [STOC 2025] showed that for any fixed k, the k-edge connected components of an undirected graph can be computed in linear time. In contrast, the directed case remains significantly more challenging: linear-time algorithms are only known for k ≤ 3, and for any fixed k > 3, the best known bound for sparse or moderately dense graphs is still the O(mn)-time algorithm of Nagamochi and Watanabe (1993).
In this paper, we break the O(mn) barrier for all k = o(n^{1/4}/√{log{n}}). We present a randomized algorithm that computes the (k+2)-edge-connected components of a k-edge-connected directed graph in O(k² m √n log n) time, for any k. This constitutes the first improvement over the classic Nagamochi-Watanabe bound for any constant k > 3. Our approach introduces new structural insights into directed edge-cuts and combines these with both new and existing techniques. A central contribution of our work is a substantial simplification and generalization of the framework introduced in [Loukas Georgiadis et al., 2023], which achieved an Õ(m√m) bound for computing the 3-edge-connected components of a digraph. In addition, we develop a variant of our algorithm that achieves the same O(m √n log n) running time for computing the 4-edge-connected components of a general directed graph.

Cite As Get BibTex

Loukas Georgiadis, Evangelos Kipouridis, Evangelos Kosinas, Charis Papadopoulos, and Nikos Parotsidis. Computing the (k+2)-Edge-Connected Components in k-Edge-Connected Digraphs in Subquadratic Time. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 95:1-95:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.95

Author Details

Loukas Georgiadis
  • Department of Computer Science & Engineering, University of Ioannina, Greece
Evangelos Kipouridis
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Evangelos Kosinas
  • Department of Computer Science & Engineering, University of Ioannina, Greece
  • Archimedes, Athena RC, Greece
Charis Papadopoulos
  • Department of Mathematics, University of Ioannina, Greece
Nikos Parotsidis
  • Google Research, Zürich, Switzerland

Funding

This work has been partially supported by project MIS 5154714 of the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGenerationEU Program.

Acknowledgements

We thank an anonymous reviewer for suggesting the use of the Picard-Queyranne graph in the proof of Proposition 9.

References

  1. Amir Abboud, Robert Krauthgamer, and Ohad Trabelsi. APMF < APSP? Gomory-Hu tree for unweighted graphs in almost-quadratic time, 2021. URL: https://arxiv.org/abs/2106.02981.
  2. Amir Abboud, Robert Krauthgamer, and Ohad Trabelsi. Subcubic algorithms for gomory–hu tree in unweighted graphs. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1725-1737, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3406325.3451073.
  3. Amir Abboud, Jason Li, Debmalya Panigrahi, and Thatchaphol Saranurak. All-pairs max-flow is no harder than single-pair max-flow: Gomory–Hu trees in almost-linear time. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 2204-2212, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00137.
  4. Shyan Akmal. An enumerative perspective on connectivity. In 2024 Symposium on Simplicity in Algorithms (SOSA), pages 179-198, 2024. URL: https://doi.org/10.1137/1.9781611977936.18.
  5. Shyan Akmal and Ce Jin. An Efficient Algorithm for All-Pairs Bounded Edge Connectivity. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:20, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.11.
  6. 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, 2025. URL: https://doi.org/10.1137/1.9781611978322.63.
  7. András A. Benczúr. Counterexamples for directed and node capacitated cut-trees. SIAM Journal on Computing, 24(3):505-510, 1995. URL: https://doi.org/10.1137/S0097539792236730.
  8. Shiri Chechik, Thomas Dueholm Hansen, Giuseppe F Italiano, Veronika Loitzenbauer, and Nikos Parotsidis. Faster algorithms for computing maximal 2-connected subgraphs in sparse directed graphs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1900-1918. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.124.
  9. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. In Proceedings of the 63rd IEEE Symposium on Foundations of Computer Science, FOCS ’22, 2022. URL: https://doi.org/10.48550/arXiv.2203.00671.
  10. Ho Yee Cheung, Lap Chi Lau, and Kai Man Leung. Graph connectivities, network coding, and expander graphs. SIAM J. Comput., 42(3):733-751, January 2013. URL: https://doi.org/10.1137/110844970.
  11. D. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, USA, 2010. Google Scholar
  12. Sebastian Forster, Danupon Nanongkai, Liu Yang, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Computing and testing small connectivity in near-linear time and queries via fast local cut algorithms. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2046-2065. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.126.
  13. H. N. Gabow. Path-based depth-first search for strong and biconnected components. Information Processing Letters, 74:107-114, 2000. URL: https://doi.org/10.1016/S0020-0190(00)00051-X.
  14. Harold N. Gabow. The minset-poset approach to representations of graph connectivity. ACM Transactions on Algorithms, 12(2):24:1-24:73, February 2016. URL: https://doi.org/10.1145/2764909.
  15. Z. Galil and G. F. Italiano. Reducing edge connectivity to vertex connectivity. SIGACT News, 22(1):57-61, March 1991. URL: https://doi.org/10.1145/122413.122416.
  16. Loukas Georgiadis, Giuseppe F. Italiano, and Evangelos Kosinas. Computing the 4-edge-connected components of a graph in linear time. In Proc. 29th European Symposium on Algorithms, pages 47:1-47:17, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.47.
  17. Loukas Georgiadis, Giuseppe F. Italiano, and Evangelos Kosinas. Computing the 3-edge-connected components of directed graphs in linear time. In 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024, Chicago, IL, USA, October 27-30, 2024, pages 62-85. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00015.
  18. Loukas Georgiadis, Giuseppe F. Italiano, Luigi Laura, and Nikos Parotsidis. 2-edge connectivity in directed graphs. ACM Trans. Algorithms, 13(1), October 2016. URL: https://doi.org/10.1145/2968448.
  19. Loukas Georgiadis, Giuseppe F. Italiano, and Nikos Parotsidis. Strong connectivity in directed graphs under failures, with applications. SIAM J. Comput., 49(5):865-926, 2020. URL: https://doi.org/10.1137/19M1258530.
  20. Loukas Georgiadis, Evangelos Kipouridis, Evangelos Kosinas, Charis Papadopoulos, and Nikos Parotsidis. Computing the (k+2)-edge-connected components in k-edge-connected digraphs in subquadratic time, 2026. URL: https://arxiv.org/abs/2604.27474.
  21. Loukas Georgiadis, Evangelos Kipouridis, Charis Papadopoulos, and Nikos Parotsidis. Faster computation of 3-edge-connected components in digraphs. In Proc. 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 2489-2531. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH96.
  22. Loukas Georgiadis and Robert E. Tarjan. Dominator tree certification and divergent spanning trees. ACM Transactions on Algorithms, 12(1):11:1-11:42, November 2015. URL: https://doi.org/10.1145/2764913.
  23. Ramesh Hariharan, Telikepalli Kavitha, and Debmalya Panigrahi. Efficient algorithms for computing all low s-t edge connectivities and related problems. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '07, pages 127-136, USA, 2007. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=1283383.1283398.
  24. M. Henzinger, S. Rao, and D. Wang. Local flow partitioning for faster edge connectivity. SIAM Journal on Computing, 49(1):1-36, 2020. URL: https://doi.org/10.1137/18M1180335.
  25. J. E. Hopcroft and R. E. Tarjan. Dividing a graph into triconnected components. SIAM Journal on Computing, 2(3):135-158, 1973. URL: https://doi.org/10.1137/0202012.
  26. Gary Hoppenworth, Thatchaphol Saranurak, and Benyu Wang. Near-optimal fault-tolerant strong connectivity preservers. In Proceedings of the 66th Annual Symposium on Foundations of Computer Science (FOCS 2025), 2025. Google Scholar
  27. Andreas Huck. Disproof of a conjecture about independent branchings in k-connected directed graphs. Journal of Graph Theory, 20(2):235-239, 1995. URL: https://doi.org/10.1002/jgt.3190200212.
  28. K.-I. Kawarabayashi and M. Thorup. Deterministic edge connectivity in near-linear time. Journal of the ACM, 66(1), December 2018. URL: https://doi.org/10.1145/3274663.
  29. Tuukka Korhonen. Linear-time algorithms for k-edge-connected components, k-lean tree decompositions, and more. In 57th ACM Symposium on Theory of Computing (STOC 2025), 2025. Google Scholar
  30. Evangelos Kosinas. Computing the 5-edge-connected components in linear time. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1887-2119, 2024. URL: https://doi.org/10.1137/1.9781611977912.76.
  31. Jason Li, Debmalya Panigrahi, and Thatchaphol Saranurak. A nearly optimal all-pairs min-cuts algorithm in simple graphs. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 1124-1134, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00111.
  32. K. Menger. Zur allgemeinen kurventheorie. Fund. Math., 10:96-115, 1927. Google Scholar
  33. W. Nadara, M. Radecki, M. Smulewicz, and M. Sokolowski. Determining 4-edge-connected components in linear time. In Proc. 29th European Symposium on Algorithms, 2021. Google Scholar
  34. H. Nagamochi and T. Ibaraki. A linear time algorithm for computing 3-edge-connected components in a multigraph. Japan J. Indust. Appl. Math, 9(163), 1992. URL: https://doi.org/10.1007/BF03167564.
  35. H. Nagamochi and T. Ibaraki. Algorithmic Aspects of Graph Connectivity. Cambridge University Press, 2008. 1st edition. Google Scholar
  36. H. Nagamochi and Toshimasa Watanabe. Computing k-edge-connected components of a multigraph (special section on discrete mathematics and its applications). IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 76:513-517, 1993. Google Scholar
  37. Danupon Nanongkai, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Breaking quadratic time for small vertex connectivity and an approximation scheme. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 241-252, 2019. URL: https://doi.org/10.1145/3313276.3316394.
  38. Jean-Claude Picard and Maurice Queyranne. On the structure of all minimum cuts in a network and applications. Math. Program., 13(1):8-16, 1980. Google Scholar
  39. Claus-Peter Schnorr. Bottlenecks and edge connectivity in unsymmetrical networks. SIAM Journal on Computing, 8(2):265-274, 1979. URL: https://doi.org/10.1137/0208019.
  40. R. E. Tarjan. Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2):146-160, 1972. URL: https://doi.org/10.1137/0201010.
  41. Y. H. Tsin. Yet another optimal algorithm for 3-edge-connectivity. Journal of Discrete Algorithms, 7(1):130-146, 2009. Selected papers from the 1st International Workshop on Similarity Search and Applications (SISAP). URL: https://doi.org/10.1016/j.jda.2008.04.003.
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