Dissipative Kerr solitons (DKS) in high-Q microresonators have attracted considerable attention for their broadband optical frequency combs and ultra-short pulse generation. Owing to thermal effects, complicated tuning strategies are required to generate and sustain the single-soliton state in microresonators. In this paper, we propose a novel microresonator scheme based on the Fabry-Pérot fiber resonator and single-layer graphene saturable absorber (SA) and demonstrate that this design allows deterministic single-soliton generation without frequency tuning and has strong robustness against pump perturbation. The soliton range and thermal instability of the proposed device are also discussed. This work facilitates a novel nonlinear platform connecting high-Q microresonators and conventional SA-assisted mode-locking operations.