We study S ubgraph I somorphism 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 S ubgraph I somorphism with respect to the forbidden minor, where the only unsettled case is $$P_{5}$$ 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 $$P_{5}$$ P 5 or the disjoint union of two $$P_{5}$$ P 5 ’s. As a byproduct, we show that S ubgraph I somorphism is fixed-parameter tractable parameterized by vertex integrity. Using similar techniques, we also observe that S ubgraph I somorphism is fixed-parameter tractable parameterized by neighborhood diversity.

中文翻译：

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排除子结构的图类上的子图同构

我们研究由固定禁止图定义的图类上的子图 I 同构。尽管有多种禁止图的方法，但我们观察到关注次要关系是合理的，因为其他众所周知的关系会导致微不足道或等效的问题。当禁止次要连接时，我们提出了关于禁止次要的子图 I 同构的复杂性的近似二分法，其中唯一未解决的情况是 $$P_{5}$$P 5 ，即五个顶点的路径。然后我们还考虑可能断开连接的被禁止的未成年人的一般情况。我们展示了固定参数易处理案例和随机 XP 时间可解决案例，这些案例由禁止的次要 H 的大小参数化。我们还表明，通过稍微概括易处理的案例，问题变得 NP 完全。所有不稳定的情况都等价于 $$P_{5}$$ P 5 或两个 $$P_{5}$$ P 5 的不相交联合。作为副产品，我们证明了子图 I 同构是由顶点完整性参数化的固定参数易处理的。使用类似的技术，我们还观察到子图 I 同构是由邻域多样性参数化的固定参数易处理的。