  We present a family of minimal modal logics (namely, modal logics based on
minimal propositional logic) corresponding each to a different classical modal
logic. The minimal modal logics are defined based on their classical
counterparts in two distinct ways: (1) via embedding into fusions of classical
modal logics through a natural extension of the G\"odel-Johansson translation
of minimal logic into modal logic S4; (2) via extension to modal logics of the
multi- vs. single-succedent correspondence of sequent calculi for classical and
minimal logic. We show that, despite being mutually independent, the two
methods turn out to be equivalent for a wide class of modal systems. Moreover,
we compare the resulting minimal version of K with the constructive modal logic
CK studied in the literature, displaying tight relations among the two systems.
Based on these relations, we also define a constructive correspondent for each
minimal system, thus obtaining a family of constructive modal logics which
includes CK as well as other constructive modal logics studied in the
literature.
