Subgraph isomorphism on graph classes that exclude a substructure

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 the path of five vertices. We then also consider the general case of possibly disconnected forbidden minors. We show in particular that: the problem is fixed-parameter tractable parameterized by the size of the forbidden minor H when H is a linear forest such that at most one component has four vertices and all other components have three or less vertices; and it is NP-complete if H contains four or more components with at least five vertices each. 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.