  Let $(X,\om)$ be a symplectic manifold and $L$ be a Lagrangian submanifold
diffeomorphic to $S^n$, $\R\P^n$, or a Lens space of a certain type. Using the
symplectic cut and symplectic sum constructions, we express the open
Gromov-Witten invariants of $(X,L)$ in terms of open Gromov-Witten invariants
of a pair $(X_-,L)$ determined by $L$ and the standard Gromov-Witten invariants
of a symplectic manifold $X_+$ determined by $(X,L)$. We also describe other
applications of this approach.
