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 $f\left(R\right)$ gravity background. To achieve that goal, we consider two different models of the $f\left(R\right)$ 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, $VelaX-1,$ $PSRJ1614-2230,$ $4U1608-52,$ $CenX-3$ and $4U1820-30.$ 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 $f\left(R\right)$ gravity models. All these physical attributes indicate the consistent behavior of our models. Our investigation also suggests that $f\left(R\right)$ theory of gravity appears as a suitable theory in describing the viability of a new classification of embedded class-I solutions of compact objects.

本文的主要目的是利用Karmarkar条件研究密星的嵌入式I类解的新分类的存在。 $F（\mathrm{[R}）$重力背景。为了实现该目标，我们考虑了两种不同的$F（\mathrm{[R}）$考虑各向异性物质分布的静态球对称时空的重力理论。此外，我们采用Karmarkar条件将度量势的两个分量联系起来$G_{\mathrm{[R}\mathrm{[R}}$ 和 $G_{ŤŤ}$。我们假设一个特定的模型用于一个度量电位，并通过Karmarkar条件获得第二个模型。此外，我们还利用这些紧凑型恒星的观测数据来计算常数参数的值，即$\mathrm{伏特}Ë升\mathrm{一种}X-\mathrm{1个}，$ $P\mathrm{小号}\mathrm{[R}Ĵ1614-2230，$ $4ü1608-52，$ $CËñX-3$ 和 $4ü1820-30。$ 我们执行不同的物理测试，例如能量密度和压力分量的变化行为，稳定性和平衡条件，能量约束，质量函数和绝热指数，以检查燃料电池的可行性。 $F（\mathrm{[R}）$重力模型。所有这些物理属性表明了我们模型的一致行为。我们的调查还表明$F（\mathrm{[R}）$ 引力论似乎是描述紧凑对象的嵌入式I类解决方案的新分类可行性的一种合适理论。