Structural Parameterizations of b-Coloring

Authors Lars Jaffke , Paloma T. Lima , Roohani Sharma

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Lars Jaffke
  • University of Bergen, Norway
Paloma T. Lima
  • IT University of Copenhagen, Denmark
Roohani Sharma
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany


We would like to thank an anonymous reviewer for carefully reading our paper and for many useful comments.

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Lars Jaffke, Paloma T. Lima, and Roohani Sharma. Structural Parameterizations of b-Coloring. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


The b-Coloring problem, which given a graph G and an integer k asks whether G has a proper k-coloring such that each color class has a vertex adjacent to all color classes except its own, is known to be FPT parameterized by the vertex cover number and XP and 𝖶[1]-hard parameterized by clique-width. Its complexity when parameterized by the treewidth of the input graph remained an open problem. We settle this question by showing that b-Coloring is XNLP-complete when parameterized by the pathwidth of the input graph. Besides determining the precise parameterized complexity of this problem, this implies that b-Coloring parameterized by pathwidth is 𝖶[t]-hard for all t, and resolves the parameterized complexity of b-Coloring parameterized by treewidth. We complement this result by showing that b-Coloring is FPT when parameterized by neighborhood diversity and by twin cover, two parameters that generalize vertex cover to more dense graphs, but are incomparable to pathwidth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
  • b-coloring
  • structural parameterization
  • XNLP
  • pathwidth
  • neighborhood diversity
  • twin cover


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  1. Davi de Andrade and Ana Silva. (Sub)fall coloring and b-coloring parameterized by treewidth. In Anais do VII Encontro de Teoria da Computação, ETC 2022, pages 69-72, 2022. Google Scholar
  2. 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, pages 3282-3304. SIAM, 2023. URL:
  3. Hans L. Bodlaender, Gunther Cornelissen, and Marieke van der Wegen. Problems hard for treewidth but easy for stable gonality. In Michael A. Bekos and Michael Kaufmann, editors, Proceedings of the 48th International Workshop Graph-Theoretic Concepts in Computer Science, WG 2022, volume 13453 of Lecture Notes in Computer Science, pages 84-97. Springer, 2022. URL:
  4. Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, and Paloma T. Lima. XNLP-completeness for parameterized problems on graphs with a linear structure. In Holger Dell and Jesper Nederlof, editors, Proceedings of the 17th International Symposium on Parameterized and Exact Computation, IPEC 2022, volume 249 of LIPIcs, pages 8:1-8:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  5. Hans L. Bodlaender, Carla Groenland, Jesper Nederlof, and Céline M. F. Swennenhuis. Parameterized problems complete for nondeterministic FPT time and logarithmic space. In Proceedings 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, pages 193-204, 2021. URL:
  6. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. Journal of the ACM, 69(1):3:1-3:46, 2022. URL:
  7. Flavia Bonomo, Guillermo Durán, Frederic Maffray, Javier Marenco, and Mario Valencia-Pabon. On the b-coloring of cographs and P₄-sparse graphs. Graphs and Combinatorics, 25(2):153-167, 2009. Google Scholar
  8. Flavia Bonomo, Oliver Schaudt, Maya Stein, and Mario Valencia-Pabon. b-Coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs. Algorithmica, 73(2):289-305, 2015. Google Scholar
  9. Victor A. Campos, Carlos V. Lima, Nicolas A. Martins, Leonardo Sampaio, Marcio C. Santos, and Ana Silva. The b-chromatic index of graphs. Discrete Mathematics, 338(11):2072-2079, 2015. Google Scholar
  10. Victor A. Campos, Cláudia Linhares-Sales, Rudini Sampaio, and Ana Karolinna Maia. Maximization coloring problems on graphs with few P₄. Discrete Applied Mathematics, 164:539-546, 2014. Google Scholar
  11. Derek G. Corneil and Udi Rotics. On the relationship between clique-width and treewidth. SIAM Journal on Computing, 34(4):825-847, 2005. URL:
  12. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Springer, 1999. Google Scholar
  13. Michael Elberfeld, Christoph Stockhusen, and Till Tantau. On the space and circuit complexity of parameterized problems: Classes and completeness. Algorithmica, 71(3):661-701, 2015. URL:
  14. Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In Gregory Gutin and Stefan Szeider, editors, Parameterized and Exact Computation, pages 163-176, Cham, 2013. Springer International Publishing. Google Scholar
  15. Robert Ganian. Twin-cover: Beyond vertex cover in parameterized algorithmics. In Dániel Marx and Peter Rossmanith, editors, Parameterized and Exact Computation, pages 259-271, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. Google Scholar
  16. Frédéric Havet, Claudia Linhares Sales, and Leonardo Sampaio. b-Coloring of tight graphs. Discrete Applied Mathematics, 160(18):2709-2715, 2012. Google Scholar
  17. Robert W. Irving and David F. Manlove. The b-chromatic number of a graph. Discrete Applied Mathematics, 91(1-3):127-141, 1999. Google Scholar
  18. Lars Jaffke, Paloma T. Lima, and Daniel Lokshtanov. b-Coloring parameterized by clique-width. Theory of Computing Systems, 2023. To appear. Conference version in STACS 2021, pages 43:1-43:15. Google Scholar
  19. Klaus Jansen and Lars Rohwedder. On integer programming and convolution. In Avrim Blum, editor, Proceedings of the 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, volume 124 of LIPIcs, pages 43:1-43:17. Schloss Dagstuhl, 2019. arxiv:1803.04744. URL:
  20. Martin Koutecký. A note on coloring (4K₁, C₄, C₆)-free graphs with a C₇. Graphs Comb., 38(5):149, 2022. URL:
  21. Jan Kratochvíl, Zsolt Tuza, and Margit Voigt. On the b-chromatic number of graphs. In WG 2002, pages 310-320, 2002. Google Scholar
  22. Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19-37, 2012. URL:
  23. Ana Shirley Ferreira da Silva. The b-chromatic number of some tree-like graphs. PhD thesis, Université Joseph-Fourier-Grenoble I, 2010. Google Scholar
  24. Clara Inés Betancur Velasquez, Flavia Bonomo, and Ivo Koch. On the b-coloring of P₄-tidy graphs. Discrete Applied Mathematics, 159(1):60-68, 2011. Google Scholar