Let $\mathcal{F}$ be a family of graphs. For a graph G, define $\theta _{F}(G)$ as the minimum number of induced subgraphs of G, each isomorphic to a member of $\mathcal{F}$ , needed to cover $V(G)$ , and $\alpha _{F}(G)$ as the maximum number of vertices in G such that no two are contained in an induced subgraph of G isomorphic to a member of $\mathcal{F}$ . In this paper, we focus on the fundamental inequality of graphs, $\theta _{F}(G) \geq \alpha _{F}(G)$ , and the characterization of $\mathcal{F}$ -perfect graphs, where equality holds for all induced subgraphs. Specifically, we investigate induced star-perfect graphs, where $\mathcal{F}$ is the family of stars and $\omega _{F}(G)$ is the size of a maximum induced star in G. The characterization of induced star-perfect graphs by a set of forbidden induced subgraphs was conjectured by Ravindra in 2011 and was proved in 2024. Here, we present a shorter proof of this characterization, applying our main result that the inequality $\alpha _{F}(H) \omega _{F}(H) \geq |V(H)|$ holds for every induced subgraph H of an induced star-perfect graph.
