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In this Letter, we use first-principles simulations to demonstrate the absence of Fermi-level pinning when graphene is in contact with transition metal dichalcogenides (TMDs). We find that formation of either an ohmic or Schottky contact is possible. Then we show that, due to the shallow density of states around its Fermi level, the work function of graphene can be tuned by ion adsorption. Finally we combine work function tuning of graphene and an ideal contact between graphene and TMDs to propose an ionic barristor design that can tune the work function of graphene with a much wider margin than current barristor designs, achieving a dynamic switching among p-type ohmic contact, Schottky contact, and n-type ohmic contact in one device.