Abstract Let F F be a family of graphs. For a graph G, define ₅ (G) θ F (G) as the minimum number of induced subgraphs of G, each isomorphic to a member of F F, needed to cover V (G) V (G), and ₅ (G) α 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 F F. In this paper, we focus on the fundamental inequality of graphs, ₅ (G) ₅ (G) θ F (G) ≥ α F (G), and the characterization of F F -perfect graphs, where equality holds for all induced subgraphs. Specifically, we investigate induced star-perfect graphs, where F F is the family of stars and ₅ (G) ω 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 ₅ (H) ₅ (H) |V (H) | α F (H) ω F (H) ≥ | V (H) | holds for every induced subgraph H of an induced star-perfect graph.
Alex et al. (2026) studied this question.