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Subgraph Isomorphism on Graph Classes that Exclude a Substructure

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Abstract

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.

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Notes

  1. We assume that the readers are familiar with the concept of parameterized complexity. See e.g. [6] for basic definitions omitted here.

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Funding

Partially supported by NETWORKS (the Networks project, funded by the Netherlands Organization for Scientific Research NWO), the ELC project (the project Exploring the Limits of Computation, funded by MEXT), JSPS/MEXT KAKENHI Grant Numbers JP24106004, JP18K11168, JP18K11169, JP18H04091, JP18H06469, JP15K00009, JST CREST Grant Number JPMJCR1402, and Kayamori Foundation of Informational Science Advancement. The authors thank Momoko Hayamizu, Kenji Kashiwabara, Hirotaka Ono, Ryuhei Uehara, and Koichi Yamazaki for helpful discussions. The authors are grateful to the anonymous reviewer of an earlier version of this paper who pointed out a gap in a proof. A preliminary version appeared in the proceedings of the 11th International Conference on Algorithms and Complexity (CIAC 2019), Lecture Notes in Computer Science 11485 (2019) 87–98.

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Authors and Affiliations

  1. Utrecht University, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. Chuo University, Bunkyo-ku, Tokyo, Japan

    Tesshu Hanaka

  3. Kyoto University, Kyoto, Japan

    Yasuaki Kobayashi & Yusuke Kobayashi

  4. The University of Electro-Communications, Chofu, Tokyo, Japan

    Yoshio Okamoto

  5. RIKEN Center for Advanced Intelligence Project, Tokyo, Japan

    Yoshio Okamoto

  6. Nagoya University, Nagoya, 464-8601, Japan

    Yota Otachi

  7. Maastricht University, Maastricht, The Netherlands

    Tom C. van der Zanden

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  1. Hans L. Bodlaender
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  2. Tesshu Hanaka
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  3. Yasuaki Kobayashi
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  4. Yusuke Kobayashi
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  5. Yoshio Okamoto
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  6. Yota Otachi
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  7. Tom C. van der Zanden
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Correspondence to Yota Otachi.

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Bodlaender, H.L., Hanaka, T., Kobayashi, Y. et al. Subgraph Isomorphism on Graph Classes that Exclude a Substructure. Algorithmica 82, 3566–3587 (2020). https://doi.org/10.1007/s00453-020-00737-z

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