  In this paper, we examine neutron star structure in perturbative $f(R)$
gravity models with realistic equation of state. We obtain mass-radius
relations in two gravity models of the form $f_{1}(R)=R+ \alpha
R(e^{-R/R_0}-1)$ and $f_{2}(R)=R+\alpha R^2$. For this purpose, we consider NS
with several nucleonic as well as strange EoSs generated in the framework of
relativistic mean field models. The strange particles in the core of NS are in
the form of $\Lambda$ hyperons and quarks, in addition to the nucleons and
leptons. The M-R relation of the chosen EoSs lies well within the observational
limit in the case of GR. We show that these EoSs provide the most stringent
constraint on the perturbative parameter $\alpha$ and therefore can be
considered as important experimental probe for modified gravity at
astrophysical level.
