  We study carrier transport through graphene on SrTiO$_3$ substrates by
considering relative contributions of Coulomb and resonant impurity scattering
to graphene resistivity. We establish that charged impurity scattering must
dominate graphene transport as the charge neutrality point is approached by
lowering the carrier density, and in the higher density regime away from the
neutrality point a dual model including both charged impurities and resonant
defects gives an excellent description of graphene transport on SrTiO$_3$
substrates. We further establish that the non-universal high-density behavior
of $\sigma(n)$ in different graphene samples on various substrates arises from
the competition among different scattering mechanisms, and it is in principle
entirely possible for graphene transport to be dominated by qualitatively
different scattering mechanisms at high and low carrier densities.
