  We investigate approximations of the vertex coupling on a star-shaped graph
by families of operators with singularly scaled rank-one interactions. We find
a family of vertex couplings, generalizing the $\delta'$-interaction on the
line, and show that with a suitable choice of the parameters they can be
approximated in this way in the norm-resolvent sense. We also analyze spectral
properties of the involved operators and demonstrate convergence of the
corresponding on-shell scattering matrices.
