A Characterization of Graphs with Semitotal Domination Number One-Third 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 \(\gamma _{t2}(G)\) ${\gamma }_{t2}(G)$ , is the minimum cardinality of a semitotal dominating set in G. Zhu et al. (Gr Combin 33, 1119–1130, 2017) proved that if \(G\notin \{K_4,N_2\}\) $G\notin \{K_4,N_2\}$ is a connected claw-free cubic graph of order n, then \(\gamma _{t2}(G)\le \frac{n}{3}\) ${\gamma }_{t2}(G)\le \frac{n}{3}$ , which is sharp. They proposed the problem of characterizing the extremal graphs. We completely solve this problem. There are ten classes of graphs, three of which are infinite families of graphs.