  We consider a family of open sets $M_\epsilon$ which shrinks with respect to
an appropriate parameter $\epsilon$ to a graph. Under the additional assumption
that the vertex neighbourhoods are small we show that the appropriately shifted
Dirichlet spectrum of $M_\epsilon$ converges to the spectrum of the
(differential) Laplacian on the graph with Dirichlet boundary conditions at the
vertices, i.e., a graph operator without coupling between different edges. The
smallness is expressed by a lower bound on the first eigenvalue of a mixed
eigenvalue problem on the vertex neighbourhood. The lower bound is given by the
first transversal mode of the edge neighbourhood. We also allow curved edges
and show that all bounded eigenvalues converge to the spectrum of a Laplacian
acting on the edge with an additional potential coming from the curvature.
