  For any particular class of graphs, algorithms for computational problems
restricted to the class often rely on structural properties that depend on the
specific problem at hand. This begs the question if a large set of such results
can be explained by some common problem conditions. We propose such conditions
for $HH$-subgraph-free graphs. For a set of graphs $HH$, a graph $G$ is
$HH$-subgraph-free if $G$ does not contain any of graph from $H$ as a subgraph.
Our conditions are easy to state. A graph problem must be efficiently solvable
on graphs of bounded treewidth, computationally hard on subcubic graphs, and
computational hardness must be preserved under edge subdivision of subcubic
graphs. Our meta-classification says that if a graph problem satisfies all
three conditions, then for every finite set $HH$, it is ``efficiently
solvable'' on $HH$-subgraph-free graphs if $HH$ contains a disjoint union of
one or more paths and subdivided claws, and is ``computationally hard''
otherwise. We illustrate the broad applicability of our meta-classification by
obtaining a dichotomy between polynomial-time solvability and NP-completeness
for many well-known partitioning, covering and packing problems, network design
problems and width parameter problems. For other problems, we obtain a
dichotomy between almost-linear-time solvability and having no
subquadratic-time algorithm (conditioned on some hardness hypotheses). The
proposed framework thus gives a simple pathway to determine the complexity of
graph problems on $HH$-subgraph-free graphs. This is confirmed even more by the
fact that along the way, we uncover and resolve several open questions from the
literature.
