  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.
