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Parameterized Complexity of Domination Problems Using Restricted Modular Partitions

Authors Manuel Lafond , Weidong Luo

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Manuel Lafond
  • Department of Computer Science, Université de Sherbrooke, Canada
Weidong Luo
  • Department of Computer Science, Université de Sherbrooke, Canada

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Manuel Lafond and Weidong Luo. Parameterized Complexity of Domination Problems Using Restricted Modular Partitions. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 61:1-61:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


For a graph class 𝒢, we define the 𝒢-modular cardinality of a graph G as the minimum size of a vertex partition of G into modules that each induces a graph in 𝒢. This generalizes other module-based graph parameters such as neighborhood diversity and iterated type partition. Moreover, if 𝒢 has bounded modular-width, the W[1]-hardness of a problem in 𝒢-modular cardinality implies hardness on modular-width, clique-width, and other related parameters. Several FPT algorithms based on modular partitions compute a solution table in each module, then combine each table into a global solution. This works well when each table has a succinct representation, but as we argue, when no such representation exists, the problem is typically W[1]-hard. We illustrate these ideas on the generic (α, β)-domination problem, which is a generalization of known domination problems such as Bounded Degree Deletion, k-Domination, and α-Domination. We show that for graph classes 𝒢 that require arbitrarily large solution tables, these problems are W[1]-hard in the 𝒢-modular cardinality, whereas they are fixed-parameter tractable when they admit succinct solution tables. This leads to several new positive and negative results for many domination problems parameterized by known and novel structural graph parameters such as clique-width, modular-width, and cluster-modular cardinality.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • modular-width
  • parameterized algorithms
  • W-hardness
  • 𝒢-modular cardinality


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  1. Davood Bakhshesha, Mohammad Farshia, and Mahdieh Hasheminezhada. A generalization of α-dominating set and its complexity. In The 46 th Annual Iranian Mathematics Conference, page 753, 2015. Google Scholar
  2. Rémy Belmonte, Fedor V. Fomin, Petr A. Golovach, and M. S. Ramanujan. Metric dimension of bounded tree-length graphs. SIAM J. Discret. Math., 31(2):1217-1243, 2017. URL:
  3. Nadja Betzler, Robert Bredereck, Rolf Niedermeier, and Johannes Uhlmann. On bounded-degree vertex deletion parameterized by treewidth. Discret. Appl. Math., 160(1-2):53-60, 2012. URL:
  4. Hans L. Bodlaender. Treewidth: Structure and algorithms. In Giuseppe Prencipe and Shmuel Zaks, editors, Structural Information and Communication Complexity, 14th International Colloquium, SIROCCO 2007, Castiglioncello, Italy, June 5-8, 2007, Proceedings, volume 4474 of Lecture Notes in Computer Science, pages 11-25. Springer, 2007. URL:
  5. Édouard Bonnet and Nidhi Purohit. Metric dimension parameterized by treewidth. Algorithmica, 83(8):2606-2633, 2021. URL:
  6. Vincent Bouchitté and Ioan Todinca. Treewidth and minimum fill-in: Grouping the minimal separators. SIAM J. Comput., 31(1):212-232, 2001. URL:
  7. Andreas Brandstädt, Van Bang Le, and Jeremy P Spinrad. Graph classes: a survey. SIAM, 1999. Google Scholar
  8. Mustapha Chellali, Odile Favaron, Adriana Hansberg, and Lutz Volkmann. k-domination and k-independence in graphs: A survey. Graphs Comb., 28(1):1-55, 2012. URL:
  9. Gennaro Cordasco, Luisa Gargano, and Adele A. Rescigno. Iterated type partitions. In Leszek Gasieniec, Ralf Klasing, and Tomasz Radzik, editors, Combinatorial Algorithms - 31st International Workshop, IWOCA 2020, Bordeaux, France, June 8-10, 2020, Proceedings, volume 12126 of Lecture Notes in Computer Science, pages 195-210. Springer, 2020. URL:
  10. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL:
  11. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL:
  12. F. Dahme, Dieter Rautenbach, and Lutz Volkmann. Some remarks on alpha-domination. Discuss. Math. Graph Theory, 24(3):423-430, 2004. URL:
  13. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL:
  14. Jean E. Dunbar, Dean G. Hoffman, Renu C. Laskar, and Lisa R. Markus. α-domination. Discret. Math., 211:11-26, 2000. URL:
  15. Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci., 410(1):53-61, 2009. URL:
  16. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Intractability of clique-width parameterizations. SIAM J. Comput., 39(5):1941-1956, 2010. URL:
  17. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Almost optimal lower bounds for problems parameterized by clique-width. SIAM J. Comput., 43(5):1541-1563, 2014. URL:
  18. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Clique-width III: hamiltonian cycle and the odd case of graph coloring. ACM Trans. Algorithms, 15(1):9:1-9:27, 2019. URL:
  19. Fedor V Fomin, Mathieu Liedloff, Pedro Montealegre, and Ioan Todinca. Algorithms parameterized by vertex cover and modular width, through potential maximal cliques. Algorithmica, 80:1146-1169, 2018. Google Scholar
  20. Fedor V. Fomin, Ioan Todinca, and Yngve Villanger. Large induced subgraphs via triangulations and CMSO. SIAM J. Comput., 44(1):54-87, 2015. URL:
  21. András Frank and Éva Tardos. An application of simultaneous diophantine approximation in combinatorial optimization. Comb., 7(1):49-65, 1987. URL:
  22. Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In Gregory Z. Gutin and Stefan Szeider, editors, Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, volume 8246 of Lecture Notes in Computer Science, pages 163-176. Springer, 2013. URL:
  23. T. Gallai. Transitiv orientierbare graphen. Acta Mathematica Academiae Scientiarum Hungaricae, 18(1-2):25-66, March 1967. URL:
  24. Robert Ganian, Fabian Klute, and Sebastian Ordyniak. On structural parameterizations of the bounded-degree vertex deletion problem. Algorithmica, 83(1):297-336, 2021. URL:
  25. Jiong Guo. A more effective linear kernelization for cluster editing. Theor. Comput. Sci., 410(8-10):718-726, 2009. URL:
  26. Michel Habib and Christophe Paul. A survey of the algorithmic aspects of modular decomposition. Comput. Sci. Rev., 4(1):41-59, 2010. URL:
  27. Bart M. P. Jansen, Jari J. H. de Kroon, and Michal Wlodarczyk. Vertex deletion parameterized by elimination distance and even less. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1757-1769. ACM, 2021. URL:
  28. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Math. Oper. Res., 12(3):415-440, 1987. URL:
  29. Dusan Knop. Partitioning graphs into induced subgraphs. Discret. Appl. Math., 272:31-42, 2020. URL:
  30. Manuel Lafond and Weidong Luo. Parameterized complexity of domination problems using restricted modular partitions. arXiv, 2023. URL:
  31. Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19-37, 2012. URL:
  32. James K. Lan and Gerard Jennhwa Chang. Algorithmic aspects of the k-domination problem in graphs. Discret. Appl. Math., 161(10-11):1513-1520, 2013. URL:
  33. Hendrik W. Lenstra-Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL:
  34. Ke Liu and Mei Lu. w-dominating set problem on graphs of bounded treewidth. arXiv, 2021. URL:
  35. Leila Saadi, Badreddine Benreguia, Chafik Arar, and Hamouma Moumen. Self-stabilizing algorithm for minimal (α, β)-dominating set. Int. J. Comput. Math. Comput. Syst. Theory, 7(2):81-94, 2022. URL:
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