A characterization of star-perfect graphs

Abstract

Motivated by Berge perfect graphs, we define star-perfect graphs and characterize them. For a finite simple graph G(V, E), let θ s ( G ) denote the minimum number of induced stars contained in G such that the union of their vertex sets is V(G), and let α s ( G ) denote the maximum number of vertices in G such that no two of them are contained in the same induced star of G. We call a graph G star-perfect if α s ( H ) = θ s ( H ) , for every induced subgraph H of G. A graph G is star-perfect if and only if G is ( C 3 , C 3 k + 1 , C 3 k + 2 ) -free, for every k ≥ 1 . A bipartite graph G is star-perfect if and only if every induced cycle in G is of length 6 k , k ≥ 1 . The minimum parameter θ s ( G ) and the maximum parameter α s ( G ) have been extensively studied in various contexts.