This work presents a stable and switchable dual-band generation from a polarization-dependent Erbium-doped fiber laser mode-locked by monolayer graphene as a saturable absorber. The ultrashort generated pulses at single band exhibit spectral (temporal) widths of 8.8 nm (305 fs) and 9 nm (384 fs) at 1536 nm and 1557 nm, respectively, while the simultaneous dual-band state ensured a spectral width of 5 and 6 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 EDF gain. We found a relative percentage error between experimental results and simulation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$22.7\%$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$13.3\%$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$9\%$</tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$8.3\%$</tex-math></inline-formula> for 1530 nm, 1556 nm, and dual-band left and right central wavelength, respectively.