Ratio of generalized parameters

Let P be a class of graphs. A P-set in a graph G is a vertex subset X such that G(X) ∈ P . We define the P-stability number of a graph G, αP(G), as the maximum cardinality of a P-set in G. AP-dominating set in a graph is a dominatingP-set. Accordingly, the P-domination number of a graph G, γP(G), is the minimum cardinality of a P-dominating set in G. For each r ≥ 1, we define a graph G to be an αP/γQ(r)-perfect graph if αP(H)/γQ(H) ≤ r for each induced subgraph H of G. We characterize all classes of αP/γQ(r)-perfect graphs in terms of forbidden induced subgraphs for all hereditary classes P and Q containing all edgeless graphs, and for all real numbers r ≥ 1. We propose a number of related open problems and conjectures.