Some Variations of Perfect Graphs

We consider (ψ_k−γ_k−1)-perfect graphs, i.e., graphs G for which ψ_k(H) = γ_k−1(H) for any induced subgraph H of G, where ψ_k and γ_k−1 are the k-path vertex cover number and the distance (k − 1)-domination number, respectively. We study (ψ_k−γ_k−1)-perfect paths, cycles and complete graphs for k ≥ 2. Moreover, we provide a complete characterisation of (ψ₂ − γ₁)- perfect graphs describing the set of its forbidden induced subgraphs and providing the explicit characterisation of the structure of graphs belonging to this family.