Single-Exponential FPT Algorithms for Enumerating Secluded ℱ-Free Subgraphs and Deleting to Scattered Graph Classes

Authors Bart M. P. Jansen , Jari J. H. de Kroon , Michał Włodarczyk

Thumbnail PDF


  • Filesize: 0.92 MB
  • 18 pages

Document Identifiers

Author Details

Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Jari J. H. de Kroon
  • Eindhoven University of Technology, The Netherlands
Michał Włodarczyk
  • University of Warsaw, Poland

Cite AsGet BibTex

Bart M. P. Jansen, Jari J. H. de Kroon, and Michał Włodarczyk. Single-Exponential FPT Algorithms for Enumerating Secluded ℱ-Free Subgraphs and Deleting to Scattered Graph Classes. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


The celebrated notion of important separators bounds the number of small (S,T)-separators in a graph which are "farthest from S" in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive, tailored to undirected graphs, that is phrased in terms of k-secluded vertex sets: sets with an open neighborhood of size at most k. In this terminology, the bound on important separators says that there are at most 4^k maximal k-secluded connected vertex sets C containing S but disjoint from T. We generalize this statement significantly: even when we demand that G[C] avoids a finite set ℱ of forbidden induced subgraphs, the number of such maximal subgraphs is 2^𝒪(k) and they can be enumerated efficiently. This enumeration algorithm allows us to make significant improvements for two problems from the literature. Our first application concerns the Connected k-Secluded ℱ-free subgraph problem, where ℱ is a finite set of forbidden induced subgraphs. Given a graph in which each vertex has a positive integer weight, the problem asks to find a maximum-weight connected k-secluded vertex set C ⊆ V(G) such that G[C] does not contain an induced subgraph isomorphic to any F ∈ ℱ. The parameterization by k is known to be solvable in triple-exponential time via the technique of recursive understanding, which we improve to single-exponential. Our second application concerns the deletion problem to scattered graph classes. A scattered graph class is defined by demanding that every connected component is contained in at least one of the prescribed graph classes Π_1, …, Π_d. The deletion problem to a scattered graph class is to find a vertex set of size at most k whose removal yields a graph from the class. We obtain a single-exponential algorithm whenever each class Π_i is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
  • fixed-parameter tractability
  • important separators
  • secluded subgraphs


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. René van Bevern, Till Fluschnik, George B. Mertzios, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs. Discret. Optim., 30:20-50, 2018. URL:
  2. René van Bevern, Till Fluschnik, and Oxana Yu. Tsidulko. Parameterized algorithms and data reduction for the short secluded s-t-path problem. Networks, 75(1):34-63, 2020. URL:
  3. Hans L. Bodlaender, Anuj Dawar, and Virginia V. Williams. EATCS-IPEC Nerode Prize 2020, 2020. URL:
  4. Nicolas Bousquet, Jean Daligault, and Stéphan Thomassé. Multicut is FPT. SIAM J. Comput., 47(1):166-207, 2018. URL:
  5. Shiri Chechik, Matthew P. Johnson, Merav Parter, and David Peleg. Secluded connectivity problems. Algorithmica, 79(3):708-741, 2017. URL:
  6. Jianer Chen, Yang Liu, and Songjian Lu. An improved parameterized algorithm for the minimum node multiway cut problem. Algorithmica, 55(1):1-13, 2009. URL:
  7. Jianer Chen, Yang Liu, Songjian Lu, Barry O'Sullivan, and Igor Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM, 55(5):21:1-21:19, 2008. URL:
  8. Rajesh Chitnis, Marek Cygan, MohammadTaghi Hajiaghayi, Marcin Pilipczuk, and Michal Pilipczuk. Designing FPT algorithms for cut problems using randomized contractions. SIAM J. Comput., 45(4):1171-1229, 2016. URL:
  9. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL:
  10. Marek Cygan, Pawel Komosa, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, Saket Saurabh, and Magnus Wahlström. Randomized contractions meet lean decompositions. ACM Trans. Algorithms, 17(1):6:1-6:30, 2021. URL:
  11. Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Minimum bisection is fixed-parameter tractable. SIAM J. Comput., 48(2):417-450, 2019. URL:
  12. Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. On multiway cut parameterized above lower bounds. ACM Trans. Comput. Theory, 5(1):3:1-3:11, 2013. URL:
  13. Huib Donkers, Bart M. P. Jansen, and Jari J. H. de Kroon. Finding k-secluded trees faster. In Proceeding of the 48th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2022, volume 13453 of Lecture Notes in Computer Science, pages 173-186. Springer, 2022. URL:
  14. Fedor V. Fomin, Petr A. Golovach, Nikolay Karpov, and Alexander S. Kulikov. Parameterized complexity of secluded connectivity problems. Theory Comput. Syst., 61(3):795-819, 2017. URL:
  15. L. R. Ford and D. R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956. URL:
  16. Robert Ganian, M. S. Ramanujan, and Stefan Szeider. Discovering archipelagos of tractability for constraint satisfaction and counting. ACM Trans. Algorithms, 13(2):29:1-29:32, 2017. URL:
  17. Petr A. Golovach, Pinar Heggernes, Paloma T. Lima, and Pedro Montealegre. Finding connected secluded subgraphs. J. Comput. Syst. Sci., 113:101-124, 2020. URL:
  18. Sylvain Guillemot. FPT algorithms for path-transversal and cycle-transversal problems. Discret. Optim., 8(1):61-71, 2011. URL:
  19. Yoichi Iwata, Magnus Wahlström, and Yuichi Yoshida. Half-integrality, LP-branching, and FPT algorithms. SIAM J. Comput., 45(4):1377-1411, 2016. URL:
  20. Yoichi Iwata, Yutaro Yamaguchi, and Yuichi Yoshida. 0/1/all CSPs, half-integral A-path packing, and linear-time FPT algorithms. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 462-473. IEEE Computer Society, 2018. URL:
  21. Ashwin Jacob, Jari J. H. de Kroon, Diptapriyo Majumdar, and Venkatesh Raman. Deletion to scattered graph classes I - case of finite number of graph classes. J. Comput. Syst. Sci., 138:103460, 2023. URL:
  22. Ashwin Jacob, Diptapriyo Majumdar, and Venkatesh Raman. Parameterized complexity of deletion to scattered graph classes. In Yixin Cao and Marcin Pilipczuk, editors, 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), volume 180 of LIPIcs, pages 18:1-18:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  23. Ashwin Jacob, Diptapriyo Majumdar, and Venkatesh Raman. Deletion to scattered graph classes II - improved FPT algorithms for deletion to pairs of graph classes. J. Comput. Syst. Sci., 136:280-301, 2023. URL:
  24. Bart M. P. Jansen, Jari J. H. de Kroon, and Michał Włodarczyk. Single-exponential fpt algorithms for enumerating secluded ℱ-free subgraphs and deleting to scattered graph classes, 2023. URL:
  25. Ken-ichi Kawarabayashi and Mikkel Thorup. The minimum k-way cut of bounded size is fixed-parameter tractable. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 160-169. IEEE Computer Society, 2011. URL:
  26. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Solving hard cut problems via flow-augmentation. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10-13, 2021, pages 149-168. SIAM, 2021. URL:
  27. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Directed flow-augmentation. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20-24, 2022, pages 938-947. ACM, 2022. URL:
  28. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. J. ACM, 67(3):16:1-16:50, 2020. URL:
  29. Daniel Lokshtanov and Dániel Marx. Clustering with local restrictions. Inf. Comput., 222:278-292, 2013. URL:
  30. Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Covering small independent sets and separators with applications to parameterized algorithms. ACM Trans. Algorithms, 16(3):32:1-32:31, 2020. URL:
  31. Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Reducing CMSO model checking to highly connected graphs. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 135:1-135:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL:
  32. Max-Jonathan Luckow and Till Fluschnik. On the computational complexity of length- and neighborhood-constrained path problems. Inf. Process. Lett., 156:105913, 2020. URL:
  33. Dániel Marx. Parameterized graph separation problems. Theor. Comput. Sci., 351(3):394-406, 2006. URL:
  34. Dániel Marx. Important separators and parameterized algorithms. In Petr Kolman and Jan Kratochvíl, editors, Graph-Theoretic Concepts in Computer Science - 37th International Workshop, WG 2011, Teplá Monastery, Czech Republic, June 21-24, 2011. Revised Papers, volume 6986 of Lecture Notes in Computer Science, pages 5-10. Springer, 2011. URL:
  35. Dániel Marx, Barry O'Sullivan, and Igor Razgon. Finding small separators in linear time via treewidth reduction. ACM Trans. Algorithms, 9(4):30:1-30:35, 2013. URL:
  36. Dániel Marx and Igor Razgon. Fixed-parameter tractability of multicut parameterized by the size of the cutset. SIAM J. Comput., 43(2):355-388, 2014. URL:
  37. Neeldhara Misra. Kernelization, Planar F-deletion. In Encyclopedia of Algorithms, pages 1033-1036. Springer, 2016. URL:
  38. Marcin Pilipczuk and Michal Ziobro. Experimental evaluation of parameterized algorithms for graph separation problems: Half-integral relaxations and matroid-based kernelization. CoRR, abs/1811.07779, 2018. URL:
  39. Igor Razgon and Barry O'Sullivan. Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci., 75(8):435-450, 2009. URL:
  40. A. Schrijver. Combinatorial Optimization - Polyhedra and Efficiency. Springer, 2003. Google Scholar
  41. Mingyu Xiao. Simple and improved parameterized algorithms for multiterminal cuts. Theory Comput. Syst., 46(4):723-736, 2010. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail