Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs

Authors Matthew Johnson , Barnaby Martin , Sukanya Pandey , Daniël Paulusma , Siani Smith , Erik Jan van Leeuwen

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Author Details

Matthew Johnson
  • Durham University, UK
Barnaby Martin
  • Durham University, UK
Sukanya Pandey
  • Utrecht University, The Netherlands
Daniël Paulusma
  • Durham University, UK
Siani Smith
  • University of Bristol, UK
  • Heilbronn Institute for Mathematical Research, Bristol, UK
Erik Jan van Leeuwen
  • Utrecht University, The Netherlands


We are grateful to Jelle Oostveen and Hans Bodlaender for useful discussions.

Cite AsGet BibTex

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. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


For any finite set ℋ = {H_1,…,H_p} of graphs, a graph is ℋ-subgraph-free if it does not contain any of H_1,…,H_p as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of ℋ-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
  • forbidden subgraphs
  • independent feedback vertex set
  • treewidth


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  1. Vladimir E. Alekseev and Dmitry V. Korobitsyn. Complexity of some problems on hereditary graph classes. Diskretnaya Matematika, 2:90-96, 1990. Google Scholar
  2. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12:308-340, 1991. Google Scholar
  3. Paul S. Bonsma. The complexity of the matching-cut problem for planar graphs and other graph classes. Journal of Graph Theory, 62:109-126, 2009. Google Scholar
  4. R. L. Brooks. On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society, 37(2):194-197, 1941. URL:
  5. Vašek Chvátal. Recognizing decomposable graphs. Journal of Graph Theory, 8:51-53, 1984. Google Scholar
  6. Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23:864-894, 1994. Google Scholar
  7. Carl Feghali, Felicia Lucke, Daniel Paulusma, and Bernard Ries. Matching cuts in graphs of high girth and h-free graphs. CoRR, 2212.12317, 2023. URL:
  8. Michael R. Garey and David S. Johnson. Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., USA, 1979. Google Scholar
  9. Michael R. Garey, David S. Johnson, and Larry J. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1:237-267, 1976. Google Scholar
  10. Petr A. Golovach and Daniël Paulusma. List coloring in the absence of two subgraphs. Discrete Applied Mathematics, 166:123-130, 2014. Google Scholar
  11. Petr A. Golovach, Daniël Paulusma, and Bernard Ries. Coloring graphs characterized by a forbidden subgraph. Discrete Applied Mathematics, 180:101-110, 2015. Google Scholar
  12. 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. CoRR, abs/2211.12887, 2022. URL:
  13. Marcin Kamiński. Max-Cut and containment relations in graphs. Theoretical Computer Science, 438:89-95, 2012. Google Scholar
  14. Frédéric Maffray and Myriam Preissmann. On the np-completeness of the k-colorability problem for triangle-free graphs. Discrete Math., 162:313-317, 1996. Google Scholar
  15. Barnaby Martin, Sukanya Pandey, Daniel Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity framework for forbidden subgraphs: When hardness is not preserved under edge subdivision. CoRR, 2022. URL:
  16. Karl Menger. Zur allgemeinen kurventheorie. Fund. Math., 10:96-115, 1927. Google Scholar
  17. Andrea Munaro. Boundary classes for graph problems involving non-local properties. Theoretical Computer Science, 692:46-71, 2017. Google Scholar
  18. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. Google Scholar
  19. Svatopluk Poljak. A note on stable sets and colorings of graphs. Commentationes Mathematicae Universitatis Carolinae, 015(2):307-309, 1974. URL:
  20. Ewald Speckenmeyer. Untersuchungen zum Feedback Vertex Set Problem in ungerichteten Graphen. PhD thesis, Paderborn, 1983. Google Scholar
  21. Yuma Tamura, Takehiro Ito, and Xiao Zhou. Algorithms for the independent feedback vertex set problem. IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 98-A(6):1179-1188, 2015. URL:
  22. Shuichi Ueno, Yoji Kajitani, and Shin'ya Gotoh. On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three. Discrete Mathematics, 72(1):355-360, 1988. URL:
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