Perfect graphs involving semitotal and semipaired domination

Abstract

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 $$\gamma _\mathrm{t2}(G)$$ is the minimum cardinality of a semitotal dominating set of G, and the semipaired domination number $$\gamma _\mathrm{pr2}(G)$$ is the minimum cardinality of a semipaired dominating set of G. For a graph without isolated vertices, the domination number $$\gamma (G)$$ , the total domination $$\gamma _t(G)$$ , and the paired domination number $$\gamma _\mathrm{pr}(G)$$ are related to the semitotal and semipaired domination numbers by the following inequalities: $$\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _t(G) \le \gamma _\mathrm{pr}(G)$$ and $$\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _\mathrm{pr2}(G) \le \gamma _\mathrm{pr}(G) \le 2\gamma (G)$$ . Given two graph parameters $$\mu $$ and $$\psi $$ related by a simple inequality $$\mu (G) \le \psi (G)$$ for every graph G having no isolated vertices, a graph is $$(\mu ,\psi )$$ -perfect if every induced subgraph H with no isolated vertices satisfies $$\mu (H) = \psi (H)$$ . Alvarado et al. (Discrete Math 338:1424–1431, 2015) consider classes of $$(\mu ,\psi )$$ -perfect graphs, where $$\mu $$ and $$\psi $$ are domination parameters including $$\gamma $$ , $$\gamma _t$$ and $$\gamma _\mathrm{pr}$$ . We study classes of perfect graphs for the possible combinations of parameters in the inequalities when $$\gamma _\mathrm{t2}$$ and $$\gamma _\mathrm{pr2}$$ are included in the mix. Our results are characterizations of several such classes in terms of their minimal forbidden induced subgraphs.