  A thorough understanding of many astrophysical phenomena associated with
compact objects requires reliable knowledge about both the equation of state
(EOS)of super-dense nuclear matter and the theory of strong-field gravity
simultaneously because of the EOS-gravity degeneracy. Currently, variations of
the neutron star (NS) mass-radius correlation from using alternative gravity
theories are much larger than those from changing the NS matter EOS within
known constraints. At least two independent observables are required to break
the EOS-gravity degeneracy. Using model EOSs for hybrid stars and a Yukawa-type
non-Newtonian gravity, we investigate both the mass-radius correlation and
pulsating frequencies of NSs. While the maximum mass of NSs increases, the
frequencies of the $f$, $p_1$, $p_2$, and $w_I$ pulsating modes are found to
decrease with the increasing strength of the Yukawa-type non-Newtonian gravity,
providing a useful reference for future determination simultaneously of both
the strong-field gravity and the supranuclear EOS by combining data of x-ray
and gravitational wave emissions of neutron stars.
