We introduce a new Gentzen-style framework of grafted hypersequents that
combines the formalism of nested sequents with that of hypersequents. To
illustrate the potential of the framework, we present novel calculi for the
modal logics K5 and KD5, as well as for extensions of the
modal logics K and KD with the axiom for shift
reflexivity. The latter of these extensions is also known as SDL+
in the context of deontic logic. All our calculi enjoy syntactic cut
elimination and can be used in backwards proof search procedures of optimal
complexity. The tableaufication of the calculi for K5 and
KD5 yields simplified prefixed tableau calculi for these logic
reminiscent of the simplified tableau system for S5, which might be
of independent interest