Perfect graphs involving semitotal and semipaired domination

Let G be a graph with vertex set V and no isolated vertices, and let S be a dominating set of V. The set S is a semitotal dominating set of G if every vertex in S is within distance 2 of another vertex of S. And, S is a semipaired dominating set of G if S can be partitioned into 2-element subsets such that the vertices in each 2-set are at most distance two apart. The semitotal domination number $$\gamma _\mathrm{t2}(G)$$ γ t 2 ( G ) is the minimum cardinality of a semitotal dominating set of G, and the semipaired domination number $$\gamma _\mathrm{pr2}(G)$$ γ pr 2 ( G ) is the minimum cardinality of a semipaired dominating set of G. For a graph without isolated vertices, the domination number $$\gamma (G)$$ γ ( G ) , the total domination $$\gamma _t(G)$$ γ t ( G ) , and the paired domination number $$\gamma _\mathrm{pr}(G)$$ γ pr ( G ) are related to the semitotal and semipaired domination numbers by the following inequalities: $$\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _t(G) \le \gamma _\mathrm{pr}(G)$$ γ ( G ) ≤ γ t 2 ( G ) ≤ γ t ( G ) ≤ γ pr ( G ) and $$\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _\mathrm{pr2}(G) \le \gamma _\mathrm{pr}(G) \le 2\gamma (G)$$ γ ( G ) ≤ γ t 2 ( G ) ≤ γ pr 2 ( G ) ≤ γ pr ( G ) ≤ 2 γ ( G ) . Given two graph parameters $$\mu $$ μ and $$\psi $$ ψ related by a simple inequality $$\mu (G) \le \psi (G)$$ μ ( G ) ≤ ψ ( G ) for every graph G having no isolated vertices, a graph is $$(\mu ,\psi )$$ ( μ , ψ ) -perfect if every induced subgraph H with no isolated vertices satisfies $$\mu (H) = \psi (H)$$ μ ( H ) = ψ ( H ) . Alvarado et al. (Discrete Math 338:1424–1431, 2015) consider classes of $$(\mu ,\psi )$$ ( μ , ψ ) -perfect graphs, where $$\mu $$ μ and $$\psi $$ ψ are domination parameters including $$\gamma $$ γ , $$\gamma _t$$ γ t and $$\gamma _\mathrm{pr}$$ γ pr . We study classes of perfect graphs for the possible combinations of parameters in the inequalities when $$\gamma _\mathrm{t2}$$ γ t 2 and $$\gamma _\mathrm{pr2}$$ γ pr 2 are included in the mix. Our results are characterizations of several such classes in terms of their minimal forbidden induced subgraphs.