  We study a set of scattering matrices of quantum graphs containing minimal
number of passbands, i.e., maximal number of zero elements. The cases of even
and odd vertex degree are considered. Using a solution of inverse scattering
problem, we reconstruct boundary conditions of scale-invariant vertex
couplings. Potential-controlled universal flat filtering properties are found
for considered types of vertex couplings. Obtained boundary conditions are
approximated by simple graphs carrying only $\delta$ potentials and inner
magnetic field.
