Document Open Access Logo

Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification

Authors Tala Eagling-Vose , Jorik Jooken , Felicia Lucke , Barnaby Martin , Daniël Paulusma

PDF

File

Thumbnail PDF
PDF
LIPIcs.WG.2026.16.pdf
  • Filesize: 1.05 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • colouring
  • forbidden subgraph
  • complexity dichotomy

Metrics

Abstract

We consider Colouring on graphs that are H-subgraph-free for some fixed graph H, which are graphs that do not contain H as a subgraph. To classify the complexity of Colouring on H-subgraph-free graphs for connected H, it remains to consider when H is a tree of maximum degree 4 with exactly one vertex of degree 4, or a tree of maximum degree 3 with at least two vertices of degree 3. We let H be a so-called subdivided "H"-graph, which is either a subdivided ℍ₀: a tree of maximum degree 4 that is a star, or a subdivided ℍ₁: a tree of maximum degree 3 with exactly two vertices of degree 3. We develop new decomposition theorems resulting in polynomial-time algorithms, and in combination with known results, fully classify all cases ℍ₀ and ℍ₁. To illustrate the wider applicability of our techniques, we also employ them to obtain similar new polynomial-time results for two other classic graph problems: Stable Cut and, in part, Feedback Vertex Set.

Cite As Get BibTex

Tala Eagling-Vose, Jorik Jooken, Felicia Lucke, Barnaby Martin, and Daniël Paulusma. Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.16

Author Details

Tala Eagling-Vose
  • Department of Computer Science, Durham University, UK
Jorik Jooken
  • Department of Computer Science, KU Leuven, Kortrijk, Belgium
Felicia Lucke
  • ENS Lyon, France
Barnaby Martin
  • Department of Computer Science, Durham University, UK
Daniël Paulusma
  • Department of Computer Science, Durham University, UK

Funding

  • Jooken, Jorik: supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO) with grant number 1222524N.
  • Lucke, Felicia: supported by SNSF Postdoc Mobility Grant 230578.
  • Martin, Barnaby: supported by Leverhulme Trust Research Project Grant RPG-2024-182.
  • Paulusma, Daniël: supported by Leverhulme Trust Research Project Grant RPG-2024-182.

References

  1. R. L. Brooks. On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society, 37(2):194-197, 1941. Google Scholar
  2. Tala Eagling-Vose, Jorik Jooken, Felicia Lucke, Barnaby Martin, and Daniël Paulusma. Colouring graphs without a subdivided H-graph: A full complexity classification. CoRR, abs/2512.09859, 2025. URL: https://doi.org/10.48550/arXiv.2512.09859.
  3. Thomas Emden-Weinert, Stefan Hougardy, and Bernd Kreuter. Uniquely colourable graphs and the hardness of colouring graphs of large girth. Combinatorics, Probability and Computing, 7:375-386, 1998. URL: http://journals.cambridge.org/action/displayAbstract?aid=46667.
  4. F. Galvin, I. Rival, and B. Sands. A Ramsey-type theorem for traceable graphs. Journal of Combinatorial Theory, Series B, 33(1):7-16, 1982. Google Scholar
  5. Michael R. Garey, David S. Johnson, and Larry J. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, pages 237-267, 1976. Google Scholar
  6. Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A survey on the computational complexity of coloring graphs with forbidden subgraphs. Journal of Graph Theory, 84:331-363, 2017. URL: https://doi.org/10.1002/JGT.22028.
  7. Petr A. Golovach, Daniël Paulusma, and Bernard Ries. Coloring graphs characterized by a forbidden subgraph. Discrete Applied Mathematics, 180:101-110, 2015. URL: https://doi.org/10.1016/J.DAM.2014.08.008.
  8. Matthew Johnson, Barnaby Martin, Jelle J. Oostveen, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity framework for forbidden subgraphs I: the framework. Algorithmica, 87:429-464, 2025. URL: https://doi.org/10.1007/S00453-024-01289-2.
  9. Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity framework for forbidden subgraphs III: When problems are tractable on subcubic graphs. Proc. MFCS 2023, LIPIcs, 272:57:1-57:15, 2023. URL: https://doi.org/10.4230/LIPIcs.MFCS.2023.57.
  10. Daniel Kobler and Udi Rotics. Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics, 126:197-221, 2003. URL: https://doi.org/10.1016/S0166-218X(02)00198-1.
  11. Daniel Král', Jan Kratochvíl, Zsolt Tuza, and Gerhard J. Woeginger. Complexity of coloring graphs without forbidden induced subgraphs. Proc. WG 2001, LNCS, 2204:254-262, 2001. URL: https://doi.org/10.1007/3-540-45477-2_23.
  12. László Lovász. Coverings and coloring of hypergraphs. Congr. Numer., VIII:3-12, 1973. Google Scholar
  13. Vadim V. Lozin and Marcin Kamiński. Coloring edges and vertices of graphs without short or long cycles. Contributions to Discrete Mathematics, 2, 2007. Google Scholar
  14. Vadim V. Lozin, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Mark H. Siggers, Siani Smith, and Erik Jan van Leeuwen. Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-graphs. Proc. ISAAC 2024, LIPIcs, 322:47:1-47:18, 2024. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2024.47.
  15. Augustine M. Moshi. Matching cutsets in graphs. Journal of Graph Theory, 13:527-536, 1989. URL: https://doi.org/10.1002/JGT.3190130502.
  16. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  17. Neil Robertson and Paul D. Seymour. Graph minors. III. planar tree-width. Journal of Combinatorial Theory, Series B, 36:49-64, 1984. URL: https://doi.org/10.1016/0095-8956(84)90013-3.
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