A note on semitotal domination in graphs

A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in $S$ is within distance $2$ of another vertex of $S$. The \emph{semitotal domination number}, $\gamma_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. The \emph{semitotal domination multisubdivision number} of a graph $G$, $msd_{\gamma_{t2}}(G)$, is the minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the semitotal domination number of $G$. In this paper, we show that $msd_{\gamma_{t2}}(G)\leq 3$ for any graph $G$ of order at least $3$, we also determine the semitotal domination multisubdivision number for some classes of graphs and characterize trees $T$ with $msd_{\gamma_{t2}}(T)=3$. On the other hand, we know that $\gamma_{t2}(G)$ is a parameter that is squeezed between domination number, $\gamma(G)$ and total domination number, $\gamma_t(G)$, so for any tree $T$, we investigate the ratios $\frac{\gamma_{t2}(T)}{\gamma(T)}$ and $\frac{\gamma_t(T)}{\gamma_{t2}(T)}$, and present the constructive characterizations of the families of trees achieving the upper bounds.