In this work, a graphene-based planar terahertz electromagnetically induced transparency-like (EIT-like) metamaterial, composed of two horizontally symmetric mono-layer graphene micro-ring resonators (MRR) and a vertical mono-layer graphene micro-strip resonator (MSR), has been proposed. The transparent window was observed in the y polarization direction. By changing the carrier mobility of the graphene, the transparent window could be opened and closed while the position of the EIT-like window remained unchanged. The physical mechanism of the EIT-like effect was explained by the distribution of the electric field on E z and the three-level $\Lambda $ -type system. The theoretical fitting results based on the Lorentz oscillator model and the S-parameter inversion method were consistent with the numerical simulation results. In addition, the performance of the metamaterial for detecting the refractive index of the surrounding medium was analyzed. Numerical simulation showed that when the FOM exceeded 8.0, the sensitivity was 1.6 THz/RIU. Finally, as the graphene carrier mobility increased, the group delay of the device increased. The delay time reached 1.49 ps, and the group index was as high as 400. The proposed design provides a feasible method for the development of optical switches, biochemical molecular detection, and slow light equipment.