Subgraph Isomorphism on Graph Classes that Exclude a Substructure

Hans L. Bodlaender, Tesshu Hanaka, Yasuaki Kobayashi, Yusuke Kobayashi, Yoshio Okamoto, Yota Otachi*, Tom C. van der Zanden

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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We study Subgraph Isomorphism on graph classes defined by a fixed forbidden graph. Although there are several ways for forbidding a graph, we observe that it is reasonable to focus on the minor relation since other well-known relations lead to either trivial or equivalent problems. When the forbidden minor is connected, we present a near dichotomy of the complexity of Subgraph Isomorphism with respect to the forbidden minor, where the only unsettled case is P-5, the path of five vertices. We then also consider the general case of possibly disconnected forbidden minors. We show fixed-parameter tractable cases and randomized XP-time solvable cases parameterized by the size of the forbidden minor H. We also show that by slightly generalizing the tractable cases, the problem becomes NP-complete. All unsettle cases are equivalent to P-5 or the disjoint union of two P-5's. As a byproduct, we show that Subgraph Isomorphism is fixed-parameter tractable parameterized by vertex integrity. Using similar techniques, we also observe that Subgraph Isomorphism is fixed-parameter tractable parameterized by neighborhood diversity.
Original languageEnglish
Pages (from-to)3566-3587
Number of pages22
Issue number12
Early online date2 Jul 2020
Publication statusPublished - Dec 2020


  • Subgraph isomorphism
  • Minor-free graphs
  • Parameterized complexity


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