▎ 摘　　要

We construct two types of multi-layer quantum graphs (Schrodinger operators on metric graphs) for which the dispersion function of wave vector and energy is proved to be a polynomial in the dispersion function of the single layer. This leads to the reducibility of the algebraic Fermi surface, at any energy, into several components. Each component contributes a set of bands to the spectrum of the graph operator. When the layers are graphene, AA-, AB-, and ABC-stacking are allowed within the same multi-layer structure. One of the tools we introduce is a surgery-type calculus for obtaining the dispersion function for a periodic quantum graph by joining two graphs together. Reducibility of the Fermi surface allows for the construction of local defects that engender bound states at energies embedded in the radiation continuum.