A fully continuous geometric center symmetric cross-shaped graphene structure is proposed. Each cross-shaped graphene unit cell is composed of a central graphene region and four completely symmetric graphene chips, where each graphene chip acts as both bright and dark modes simultaneously, while the central graphene region always acts as the bright mode. Through destructive interference, the structure can realize the single plasmon-induced transparency (PIT) phenomenon, where the optical responses are independent of the polarization direction of the linearly polarized light due to the symmetry of the structure. Combining numerical simulations with coupled mode theory (CMT) calculations, the modulation of the Fermi energy of graphene to the optical spectra is investigated. The results show that the spectra are blue shifted as the Fermi energy increases, and the absorption of the two absorption peaks is basically equal (48.7%) when the Fermi energy increases to 0.667 eV. Theoretical calculations show that the slow light performance of the designed structure enhances with the increase of Fermi energy, where the maximum group index is high up to 424.73. Furthermore, it is worth noting that the electrode can be made very small due to its fully continuous structure. This work provides guidance in terms of terahertz modulators, tunable absorbers, and slow light devices.