iγ(1)-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 graphs in πτ(k) are called πτ(k)-perfect. This new concept was recently introduced by I.E. Zverovich (J. Graph Theory 32 (1999) 303–310) for the domination number γ, the independent domination number i and the independence number α. He gave characterizations of the classes iα(k) and γα(k). It is a natural question arising from his work to study the class iγ(k) which generalizes the well-known domination perfect graphs. In this note we prove a sufficient condition for a graph to belong to iγ(1) and characterize all forests in iγ(1).