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

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arXiv · 1905.10670

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

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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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Hans L. Bodlaender, Tesshu Hanaka, Yasuaki Kobayashi, Yusuke Kobayashi, Yoshio Okamoto, Yota Otachi, Tom C. van der Zanden. 2019-05-25. Subgraph Isomorphism on Graph Classes that Exclude a Substructure. https://arxiv.org/abs/1905.10670

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