A characterization of Γα(k)-perfect graphs

Let π and τ be two arbitrary graph parameters that satisfy π(G)⩾τ(G) for every graph G. For any k∈N0 the class πτ(k) is the hereditary class of graphs that consists of all graphs G such that π(H)−τ(H)⩽k for every induced subgraph H of G. The elements in πτ(k) are called πτ(k)-perfect graphs. This new concept was recently introduced and studied by Zverovich in (J. Graph Theory 32 (1999) 303–310) for the domination number γ, the independent domination number i and the independence number α. Let Γ and IR denote the upper domination number and the upper irredundance number, respectively. Our main aim in this paper is the characterization of Γα(k) in terms of forbidden induced subgraphs which generalizes a recent result of Gutin and Zverovich in (Discrete Math. 190 (1998) 95–105) on upper domination perfect graphs, i.e., graphs in Γα(0). Furthermore, we extend a number of known results on the classes IRΓ(0) and Γα(0) to the classes IRΓ(k) and Γα(k).