  This paper's main aim is to investigate the existence of a new classification
of embedded class-I solutions of compact stars, by using Karmarkar condition in
$f(R)$ gravity background. To achieve that goal, we consider two different
models of the $f(R)$ theory of gravity for the static spherically symmetric
spacetime by considering anisotropic matter distribution. Further, we employ
Karmarkar condition to relate the two components of metric potentials $g_{rr}$
and $g_{tt}$. We assume a particular model for one metric potential and obtain
the second one by Karmarkar condition. Moreover, we also calculate the values
of constant parameters by using the observational data of these compact stars,
namely, $Vela~ X-1,$ $PSR~J1614-2230,$ $4U~1608-52,$ $Cen~X-3$ and
$4U~1820-30.$ We perform different physical tests like variational behavior of
energy density and pressure components, stability and equilibrium conditions,
energy constraints, mass function and adiabatic index to check the viability of
$f(R)$ gravity models. All these physical attributes indicate the consistent
behavior of our models. Our investigation also suggests that $f(R)$ theory of
gravity appears as a suitable theory in describing the viability of a new
classification of embedded class-I solutions of compact objects.
