  We present a general framework for studying the relativistic star
perturbations on a static and spherically symmetric background in full
Horndeski theories. We take a perfect fluid into account as a form of the
Schutz-Sorkin action. Our formulation is sufficiently versatile in that the
second-order actions of perturbations in odd- and even-parity sectors are
derived without choosing particular gauge conditions, so they can be used for
any convenient gauges at hand. The odd-parity sector contains one dynamical
gravitational degree of freedom coupled to a time-independent four velocity of
the fluid. In the even-parity sector there are three dynamical perturbations
associated with gravity, scalar field, and matter sectors, whose equations of
motion are decoupled from other nondynamical perturbations. For high radial and
angular momentum modes, we obtain the propagation speeds of all dynamical
perturbations and show that the perfect fluid in the even-parity sector has a
standard sound speed affected by neither gravity nor the scalar field. Our
general stability conditions and perturbation equations of motion can be
directly applied to the stabilities of neutron stars and black holes as well as
the calculations of their quasi-normal frequencies.
