This paper generalizes the results in Aswal et al. (Econ Theory 22:45–62, 2003) on dictatorial domains. This is done in two ways. In the first, the notion of connections between pairs of alternatives in Aswal et al. (2003) is weakened to weak connectedness. This notion requires the specification of four preference orderings for every alternative pair. Domains that are linked in the sense of Aswal et al. (2003) with weak connectedness replacing connectedness, are shown to be dictatorial. In the second, the notion of connections for alternative pairs is strengthened relative to its counterpart in Aswal et al. (2003). However, a domain is shown to be dictatorial if the induced graph is merely connected rather than linked. This result generalizes the result in Sato (Rev Econ Design 14:331–342, 2010) on circular domains.