  A recent paper investigated minimal R$\nu$MDM models with the type T1-iii and
T3 one-loop topologies. However, the candidate most-minimal model does not
possess an accidental symmetry - the scalar potential contains an explicit
symmetry breaking term, rendering the dark matter unstable. We present two
models that cure this problem. However, we further show that all of the
proposed minimal one-loop R$\nu$MDM models suffer from a second problem - an
additional source of explicit $Z_2$ symmetry breaking in the Yukawa sector. We
perform a more-general analysis to show that neutrino mass models using either
the type T3 or type T1-iii one-loop topologies do not give viable minimal dark
matter candidates. Consequently, one-loop models of neutrino mass with minimal
dark matter do not appear possible. Thus, presently there remains a single
known (three-loop) model of neutrino mass that gives stable dark matter without
invoking any new symmetries.
