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Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-Graphs

Authors Vadim Lozin , Barnaby Martin , Sukanya Pandey , Daniël Paulusma , Mark Siggers , Siani Smith , Erik Jan van Leeuwen

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Author Details

Vadim Lozin
  • University of Warwick, Coventry, UK
Barnaby Martin
  • Durham University, UK
Sukanya Pandey
  • Utrecht University, The Netherlands
Daniël Paulusma
  • Durham University, UK
Mark Siggers
  • Kyungpook National University, Daegu, Republic of Korea
Siani Smith
  • University of Bristol, UK
  • Heilbronn Institute for Mathematical Research, Germany
Erik Jan van Leeuwen
  • Utrecht University, The Netherlands

Acknowledgements

We are grateful to Matthew Johnson, Jelle J. Oostveen and Hans Bodlaender for useful discussions.

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Vadim Lozin, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Mark Siggers, Siani Smith, and Erik Jan van Leeuwen. Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.47

Abstract

For a fixed set H of graphs, a graph G is H-subgraph-free if G does not contain any H ∈ H as a (not necessarily induced) subgraph. A recent framework gives a complete classification on H-subgraph-free graphs (for finite sets H) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity in H-subgraph-free graphs is unknown. We study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: Hamilton Cycle, k-Induced Disjoint Paths, C₅-Colouring and Star 3-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and also from problems that do satisfy all three conditions of the framework, in particular when we forbid certain subdivisions of the "H"-graph (the graph that looks like the letter "H"). Hence, we exhibit a rich complexity landscape among problems for H-subgraph-free graph classes.

Subject Classification

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

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