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We demonstrate an experimental study on the influence of the parameters of a graphene-based saturable absorber (SA) on the performance of mode-locked Er- and Tm-doped fiber lasers. We have fabricated a set of saturable absorbers with different number of graphene layers: 9, 12, 24, 37 and 48. Each SA was characterized in terms of nonlinear optical parameters (modulation depth, saturation intensity, saturation fluence) and tested in two state-of-the-art, low-power Er- and Tm-doped fiber lasers. Our results show, that in the Er-laser the broadest output spectrum (11 nm) and shortest pulses (345 fs) are generated using 37 layers of graphene in the SA. In case of a Tm-laser, the best performance (737 fs pulses with 5.82 nm bandwidth) was achieved with 24 layers. Additionally, we show that the modulation depth of a 9-layer SA is insufficient to initiate mode-locking in both lasers. This is the first reported …