  The post-merger gravitational wave (GW) emission from a binary neutron star
merger is expected to provide exciting new constraints on the dense-matter
equation of state (EoS). Such constraints rely, by and large, on the existence
of quasi-universal relations, which relate the peak frequencies of the
post-merger GW spectrum to properties of the neutron star structure in a
model-independent way. In this work, we report on violations of existing
quasi-universal relations between the peak spectral frequency, f_2, and the
stellar radius, for EoSs models with backwards-bending slopes in their
mass-radius relations (such that the radius increases at high masses). The
violations are extreme, with variations in f_2 of up to ~600 Hz between EoSs
that predict the same radius for a 1.4 Msun neutron star, but that have
significantly different radii at higher masses. Quasi-universality can be
restored by adding in a second parameter to the fitting formulae that depends
on the slope of the mass-radius curve. We further find strong evidence that
quasi-universality is better maintained for the radii of very massive stars
(with masses 2 Msun). Both statements imply that f_2 is mainly sensitive to the
high-density EoS. Combined with observations of the binary neutron star
inspiral, these generalized quasi-universal relations can be used to
simultaneously infer the characteristic radius and slope of the neutron star
mass-radius relation.
