  We study strong instability (by blow-up) of the standing waves for the
nonlinear Schr\"odinger equation with $\delta$-interaction on a star graph
$\Gamma$. The key ingredient is a novel variational technique applied to the
standing wave solutions being minimizers of a specific variational problem. We
also show well-posedness of the corresponding Cauchy problem in the domain of
the self-adjoint operator which defines $\delta$-interaction. This permits to
prove virial identity for the $H^1$- solutions to the Cauchy problem. We also
prove certain strong instability results for the standing waves of the
NLS-$\delta'$ equation on the line.
