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.