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We propose and demonstrate multiwavelength dissipative soliton (DS) generation in an all-normal-dispersion ytterbium-doped fiber laser based on a graphene-deposited tapered fiber (GDTF) device. Due to the interaction of the graphene with the evanescent field on the taper, the GDTF device possesses the characteristics of both saturable absorption and polarizing effect. Therefore, this device not only initiates the mode-locking operation based on the saturable absorption, but also induces both the special spectral filtering and nonlinear polarization evolution for shaping pulses into DSs. Simultaneous triple-wavelength DS operation around 1035 nm is thus achieved with a pulse energy of 6.4 nJ and pulse duration of 74.6 ps. Moreover, the DS operation is very stable with an RF signal-to-noise ratio of 62.5 dB.