Restricted domains over voter preferences have been extensively studied within the area of computational social choice, initially for preferences that are total orders over the set of alternatives and subsequently for preferences that are dichotomous—i.e., that correspond to approved and disapproved alternatives. This paper contributes to the latter stream of work in a twofold manner. First, we obtain forbidden subprofile characterisations for various important dichotomous domains. Then, we are concerned with incomplete profiles that may arise in many real-world scenarios, where we have partial information about the voters’ preferences. We tackle the problem of determining whether an incomplete profile admits a completion within a certain restricted domain and design constructive, polynomial algorithms to that effect.