  We discuss the limit of small width for the Laplacian defined on a waveguide
with Robin boundary conditions. Under suitable hypothesis on the scaling of the
curvature, we prove the convergence of the Robin Laplacian to the Laplacian on
the corresponding graph. We show that the projections on each transverse mode
generically give rise to decoupling conditions between the edges of the graph
while exceptionally a coupling can occur. The non decoupling conditions are
related to the existence of resonances at the thresholds of the continuum
spectrum.
