We identify a condition on preference domains that ensures that every locally strategy-proof and unanimous random social choice function is also strategy-proof. Furthermore every unanimous, locally strategy-proof deterministic social choice function is also group strategy-proof. The condition identified is significantly weaker than the characterization condition for local-global equivalence without unanimity in Kumar et al. (2020). The condition is not necessary for equivalence with unanimous random/deterministic social choice functions. However, we show the weaker condition of connectedness remains necessary.