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Let G be a graph with no isolated vertex. A semitotal forcing set of G is a (zero) forcing set S such that every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number $F_{t2}(G)$ is the minimum cardinality of a semitotal forcing set in G. In this paper, we prove that it is NP-complete to determine the semitotal forcing number of a graph. We also prove that if $G\neq K_n$ is a connected graph of order $n\geq 4$ with maximum degree $\Delta \geq 2$ , then $F_{t2}(G)\leq (\Delta-1)n/\Delta$ , with equality if and only if either $G=C_{4}$ or $G=P_{4}$ or $G=K_{\Delta ,\Delta }$ .

For an isolate-free graph G = (V (G), E(G)), a set S ⊆ V (G) is called a semitotal forcing set of G if it is a forcing set (or a zero forcing set) of G and every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number F t2 (G) is the minimum cardinality of a semitotal forcing set in G. In this paper, we prove that if G = K 4 is a connected claw-free cubic graph of order n, then F t2 (G) ≤ 3 8 n + 1. The graphs achieving equality in this bound are characterized, an infinite set of graphs.

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