  For any finite set $\mathcal{H} = \{H_1,\ldots,H_p\}$ of graphs, a graph is
$\mathcal{H}$-subgraph-free if it does not contain any of $H_1,\ldots,H_p$ as a
subgraph. In recent work, meta-classifications have been studied: these show
that if graph problems satisfy certain prescribed conditions, their complexity
is determined on classes of $\mathcal{H}$-subgraph-free graphs. We continue
this work and focus on problems that have polynomial-time solutions on classes
that have bounded treewidth or maximum degree at most~$3$ and examine their
complexity on $H$-subgraph-free graph classes where $H$ is a connected graph.
With this approach, we obtain comprehensive classifications for (Independent)
Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This
resolves a number of open problems.
  We highlight that, to establish that Independent Feedback Vertex Set belongs
to this collection of problems, we first show that it can be solved in
polynomial time on graphs of maximum degree $3$. We demonstrate that, with the
exception of the complete graph on four vertices, each graph in this class has
a minimum size feedback vertex set that is also an independent set.
