  Given a Counting Monadic Second Order (CMSO) sentence $\psi$, the
CMSO$[\psi]$ problem is defined as follows. The input to CMSO$[\psi]$ is a
graph $G$, and the objective is to determine whether $G\models \psi$. Our main
theorem states that for every CMSO sentence $\psi$, if CMSO$[\psi]$ is solvable
in polynomial time on "globally highly connected graphs", then CMSO$[\psi]$ is
solvable in polynomial time (on general graphs). We demonstrate the utility of
our theorem in the design of parameterized algorithms. Specifically we show
that technical problem-specific ingredients of a powerful method for designing
parameterized algorithms, recursive understanding, can be replaced by a
black-box invocation of our main theorem. We also show that our theorem can be
easily deployed to show fixed parameterized tractability of a wide range of
problems, where the input is a graph $G$ and the task is to find a connected
induced subgraph of $G$ such that "few" vertices in this subgraph have
neighbors outside the subgraph, and additionally the subgraph has a
CMSO-definable property.
