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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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