In an isolate-free graph G, a subset S of vertices 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 semitotal domination number of G, denoted by $\gamma _{t2}(G)$, is the minimum cardinality of a semitotal dominating set in G. Goddard, Henning and McPillan [‘Semitotal domination in graphs’, Utilitas Math. 94 (2014), 67–81] characterised the trees and graphs of minimum degree 2 with semitotal domination number half their order. In this paper, we characterise all graphs whose semitotal domination number is half their order.

In an isolate-free graph G, a subset S of vertices 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 semitotal domination number of G, denoted by $\gamma _{t2}(G)$, is the minimum cardinality of a semitotal dominating set in G. Using edge weighting functions on semitotal dominating sets, we prove that if $G\neq N_2$ is a connected claw-free graph of order $n\geq 6$ with minimum degree $\delta (G)\geq 3$, then $\gamma _{t2}(G)\leq \frac{4}{11}n$ and this bound is sharp, disproving the conjecture proposed by Zhu et al. [‘Semitotal domination in claw-free cubic graphs’, Graphs Combin. 33(5) (2017), 1119–1130].