A characterization of star-perfect graphs

Motivated by Berge perfect graphs, we define star-perfect graphs and characterize them. For a finite simple graph G(V, E), let theta 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 alpha 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 alpha s(H)=theta s(H) , for every induced subgraph H of G. A graph G is star-perfect if and only if G is (C3,C3k+1,C3k+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 6k,k >= 1 . The minimum parameter theta s(G) and the maximum parameter alpha s(G) have been extensively studied in various contexts.