This study proposes a multifunctional absorber in the terahertz (THz) regime based on vanadium dioxide (VO2) and graphene with either-or band selector applications, which can be realized by electrically and thermally controlling the Fermi energy level of graphene and vanadium dioxide, respectively. The broadband absorption can be achieved with absorptance exceeding 90%, when the VO2 film is in the metallic phase and the Fermi energy levels of the upper and lower graphene layers are simultaneously set to 0.6 and 0 eV, respectively. The double narrowband can be realized when the VO2 film is in the insulating phase and the Fermi energy levels in upper and lower graphene layers are set as 0 and 0.8 eV, respectively. By flexibly shifting between the broadband and the double narrowband, the proposed absorber can be used as an either-or band selector, corresponding optional bandwidth from 2.05 to 2.35 THz, and 3.25 to 3.6 THz. Furthermore, single narrowband absorption can be achieved by setting the conductivity of the VO2 film to appropriate values. The proposed absorber can be used in the THz regime in applications such as multifunctional devices, switches, cloaking objects, and band selectors.