This work presents a study of dual-band generation from a polarization-dependent Erbium-doped fiber laser mode-locked by monolayer graphene as a saturable absorber. The generated pulses at single band, exhibit spectral widths of 8.5 and 8.9 nm at 1536 and 1557 nm, respectively, while the simultaneous dual-band state ensured a spectral width of 5 and 5.8 nm for both wavelengths, which are the best dual-band mode-locked performances of the literature obtained from an EDFL. Also, its dual-band spectral evolution was analyzed by changing the pumping power and the intracavity polarization state via a motorized polarization controller. Finally, our simulation provided an excellent qualitative explanation of the laser switching as a function of pumping power due to the change in erbium gain, which also explains the appearance of the dual-band. By comparing our experimental results with simulations, we obtained a relative error in spectral width around 0.2.