Document Open Access Logo

Finding b-Colorings Using Feedback Edges

Author Jakub Balabán

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2026.36.pdf
  • Filesize: 1 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • b-coloring
  • fixed-parameter algorithms
  • feedback edge number
  • distance to co-cluster

Metrics

Abstract

A b-coloring of a graph is a proper vertex coloring such that each color class contains a vertex that sees all other colors in its neighborhood. The b-coloring problem, in which the task is to decide whether a graph admits a b-coloring with k colors, is NP-complete in general but polytime solvable on trees. Moreover, it is known that b-coloring is in XP but W[t]-hard for all t ∈ ℕ when parameterized by tree-width. In fact, only very few parameters, such as the vertex cover number, were known to admit an FPT algorithm for b-coloring.
In this paper, we consider a more restrictive parameter measuring similarity to trees than tree-width, namely the feedback edge number, and show that b-coloring is fixed-parameter tractable under this parameterization. Our algorithm combines standard techniques used in parameterized algorithmics with the problem-specific ideas used in the polytime algorithm for trees. In addition, we present an FPT algorithm for b-coloring parameterized by distance to co-cluster, which is a parameter measuring similarity to complete multipartite graphs. Finally, we make several observations based on known results, including that b-coloring is W[1]-hard when parameterized by tree-depth.

Cite As Get BibTex

Jakub Balabán. Finding b-Colorings Using Feedback Edges. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.36

Author Details

Jakub Balabán
  • Faculty of Informatics, Masaryk University, Brno, Czechia

Funding

Supported by the project 26-21334S of the Czech Science Foundation.
  • Balabán, Jakub: Brno Ph.D. Talent Scholarship Holder – Funded by the Brno City Municipality

References

  1. Davi Andrade and Ana Silva. (sub)fall coloring and b-coloring parameterized by treewidth. In Anais do VII Encontro de Teoria da Computação, pages 69-72, Porto Alegre, RS, Brasil, 2022. SBC. URL: https://doi.org/10.5753/etc.2022.222982.
  2. Jakub Balabán. Finding b-colorings using feedback edges. CoRR, abs/2512.14390, 2025. URL: https://doi.org/10.48550/arXiv.2512.14390.
  3. Jakub Balabán, Robert Ganian, and Mathis Rocton. Computing twin-width parameterized by the feedback edge number and vertex integrity. SIAM J. Discret. Math., 39(4):2102-2141, 2025. URL: https://doi.org/10.1137/24M1705329.
  4. Michael J. Bannister, Sergio Cabello, and David Eppstein. Parameterized complexity of 1-planarity. J. Graph Algorithms Appl., 22(1):23-49, 2018. URL: https://doi.org/10.7155/JGAA.00457.
  5. Rémy Belmonte, Pim van 't Hof, Marcin Kaminski, Daniël Paulusma, and Dimitrios M. Thilikos. Characterizing graphs of small carving-width. Discret. Appl. Math., 161(13-14):1888-1893, 2013. URL: https://doi.org/10.1016/J.DAM.2013.02.036.
  6. Flavia Bonomo, Guillermo Durán, Frédéric Maffray, Javier Marenco, and Mario Valencia-Pabon. On the b-coloring of cographs and P_4-sparse graphs. Graphs Comb., 25(2):153-167, 2009. URL: https://doi.org/10.1007/S00373-008-0829-1.
  7. 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. URL: https://doi.org/10.1007/S00453-014-9921-5.
  8. Anudhyan Boral, Marek Cygan, Tomasz Kociumaka, and Marcin Pilipczuk. A fast branching algorithm for cluster vertex deletion. Theory of Computing Systems, 58(2):357-376, 2016. URL: https://doi.org/10.1007/S00224-015-9631-7.
  9. Victor A. Campos, Carlos Vinícius G. C. Lima, Nicolas Almeida Martins, Leonardo Sampaio Rocha, Marcio Costa Santos, and Ana Silva. The b-chromatic index of graphs. Discret. Math., 338(11):2072-2079, 2015. URL: https://doi.org/10.1016/J.DISC.2015.04.026.
  10. Victor A. Campos, Carlos Vinícius G. C. Lima, and Ana Silva. Graphs of girth at least 7 have high b-chromatic number. Eur. J. Comb., 48:154-164, 2015. URL: https://doi.org/10.1016/J.EJC.2015.02.017.
  11. Victor A. Campos, Cláudia Linhares Sales, Frédéric Maffray, and Ana Silva. b-chromatic number of cacti. Electron. Notes Discret. Math., 35:281-286, 2009. URL: https://doi.org/10.1016/J.ENDM.2009.11.046.
  12. Victor A. Campos, Cláudia Linhares Sales, Rudini Menezes Sampaio, and Ana Karolinna Maia. Maximization coloring problems on graphs with few p_4. Discret. Appl. Math., 164:539-546, 2014. URL: https://doi.org/10.1016/J.DAM.2013.10.031.
  13. Sylvie Corteel, Mario Valencia-Pabon, and Juan-Carlos Vera. On approximating the b-chromatic number. Discrete Applied Mathematics, 146(1):106-110, 2005. URL: https://doi.org/10.1016/J.DAM.2004.09.006.
  14. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  15. Josep Díaz, Jordi Petit, and Maria Serna. A survey of graph layout problems. ACM Computing Surveys (CSUR), 34(3):313-356, 2002. URL: https://doi.org/10.1145/568522.568523.
  16. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  17. Pål Grønås Drange, Markus S. Dregi, and Pim van 't Hof. On the computational complexity of vertex integrity and component order connectivity. Algorithmica, 76(4):1181-1202, 2016. URL: https://doi.org/10.1007/S00453-016-0127-X.
  18. Esther Galby, Liana Khazaliya, Fionn Mc Inerney, Roohani Sharma, and Prafullkumar Tale. Metric dimension parameterized by feedback vertex set and other structural parameters. SIAM Journal on Discrete Mathematics, 37(4):2241-2264, 2023. URL: https://doi.org/10.1137/22M1510911.
  19. Robert Ganian and Sebastian Ordyniak. The power of cut-based parameters for computing edge-disjoint paths. Algorithmica, 83(2):726-752, 2021. URL: https://doi.org/10.1007/S00453-020-00772-W.
  20. Tesshu Hanaka, Michael Lampis, Manolis Vasilakis, and Kanae Yoshiwatari. Parameterized Vertex Integrity Revisited. In Rastislav Královič and Antonín Kučera, editors, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024), volume 306 of Leibniz International Proceedings in Informatics (LIPIcs), pages 58:1-58:14, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2024.58.
  21. Frédéric Havet, Cláudia Linhares Sales, and Leonardo Sampaio Rocha. b-coloring of tight graphs. Discret. Appl. Math., 160(18):2709-2715, 2012. URL: https://doi.org/10.1016/J.DAM.2011.10.017.
  22. Robert W. Irving and David F. Manlove. The b-chromatic number of a graph. Discret. Appl. Math., 91(1-3):127-141, 1999. URL: https://doi.org/10.1016/S0166-218X(98)00146-2.
  23. Lars Jaffke, Paloma T. Lima, and Daniel Lokshtanov. b-coloring parameterized by clique-width. Theory Comput. Syst., 68(4):1049-1081, 2024. URL: https://doi.org/10.1007/S00224-023-10132-0.
  24. Lars Jaffke, Paloma T. Lima, and Roohani Sharma. Structural parameterizations of b-coloring. In Satoru Iwata and Naonori Kakimura, editors, 34th International Symposium on Algorithms and Computation, ISAAC 2023, December 3-6, 2023, Kyoto, Japan, volume 283 of LIPIcs, pages 40:1-40:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2023.40.
  25. Marko Jakovac and Sandi Klavžar. The b-chromatic number of cubic graphs. Graphs and Combinatorics, 26(1):107-118, 2010. URL: https://doi.org/10.1007/S00373-010-0898-9.
  26. Marko Jakovac and Iztok Peterin. The b-chromatic number and related topics - A survey. Discret. Appl. Math., 235:184-201, 2018. URL: https://doi.org/10.1016/J.DAM.2017.08.008.
  27. Jack de Jong. Analysing flow-like problems parameterised by tree-depth. Master’s thesis, Utrecht University, 2024. Google Scholar
  28. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  29. Christian Komusiewicz, Dieter Kratsch, and Van Bang Le. Matching cut: Kernelization, single-exponential time fpt, and exact exponential algorithms. Discret. Appl. Math., 283:44-58, 2020. URL: https://doi.org/10.1016/J.DAM.2019.12.010.
  30. Mekkia Kouider and Manouchehr Zaker. Bounds for the b-chromatic number of some families of graphs. Discrete Mathematics, 306(7):617-623, 2006. URL: https://doi.org/10.1016/J.DISC.2006.01.012.
  31. Jan Kratochvíl, Zsolt Tuza, and Margit Voigt. On the b-chromatic number of graphs. In Ludek Kucera, editor, Graph-Theoretic Concepts in Computer Science, 28th International Workshop, WG 2002, Cesky Krumlov, Czech Republic, June 13-15, 2002, Revised Papers, volume 2573 of Lecture Notes in Computer Science, pages 310-320. Springer, 2002. URL: https://doi.org/10.1007/3-540-36379-3_27.
  32. Rafael A Melo, Michell F Queiroz, and Marcio C Santos. A matheuristic approach for the b-coloring problem using integer programming and a multi-start multi-greedy randomized metaheuristic. European Journal of Operational Research, 295(1):66-81, 2021. URL: https://doi.org/10.1016/J.EJOR.2021.02.049.
  33. I. Vinod Reddy. On structural parameterizations of load coloring. Discrete Applied Mathematics, 342:181-189, 2024. URL: https://doi.org/10.1016/j.dam.2023.09.001.
  34. Paul D. Seymour and Robin Thomas. Call routing and the ratcatcher. Combinatorica, 14(2):217-241, 1994. URL: https://doi.org/10.1007/BF01215352.
  35. Ana Shirley Ferreira Da Silva. The b-chromatic number of some tree-like graphs. (Le nombre b-chromatique de quelques classes de graphes généralisant les arbres). PhD thesis, Joseph Fourier University, Grenoble, France, 2010. URL: https://tel.archives-ouvertes.fr/tel-00544757.
  36. Dimitrios M. Thilikos, Maria J. Serna, and Hans L. Bodlaender. Constructive linear time algorithms for small cutwidth and carving-width. In D. T. Lee and Shang-Hua Teng, editors, Algorithms and Computation, 11th International Conference, ISAAC 2000, Taipei, Taiwan, December 18-20, 2000, Proceedings, volume 1969 of Lecture Notes in Computer Science, pages 192-203. Springer, 2000. URL: https://doi.org/10.1007/3-540-40996-3_17.
  37. Johannes Uhlmann and Mathias Weller. Two-layer planarization parameterized by feedback edge set. Theor. Comput. Sci., 494:99-111, 2013. URL: https://doi.org/10.1016/J.TCS.2013.01.029.
  38. Clara Inés Betancur Velasquez, Flavia Bonomo, and Ivo Koch. On the b-coloring of p_4-tidy graphs. Discret. Appl. Math., 159(1):60-68, 2011. URL: https://doi.org/10.1016/J.DAM.2010.10.002.
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