Abstract Here we analyze the propagation of Dyakonov surface waves(DSWs) at the interface between isotropic dielectric material and graphene-hBN hyperstructure (GhH) whose physical properties and optical topological transition regions can be controlled by varying the Fermi energy level of graphene sheets via electrostatic biasing. Various properties of the DSWs supported at the interface have been theoretically investigated, including the confinement factor and the polarization hybridization factor. By analyzing the electric-field intensity distribution of the DSWs excited by the linearly polarized dipole, it is found that the propagation direction of the DSWs can be controlled by the chemical potential of graphene. It is shown that the GhH supporting the DSWs with high wavenumber allows the enhanced amplitude of the spontaneous emission rate (SER) for one emitters to be flexibly controlled by the voltage applied to graphene.