Geodesic
Semitotal forcing in claw-free cubic graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1373-1393

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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)≤38n+1. The graphs achieving equality in this bound are characterized, an infinite set of graphs.
Keywords: semitotal forcing number, claw-free, cubic 
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Liang, Yi-Ping; Chen, Jie; Xu, Shou-Jun. Semitotal forcing in claw-free cubic graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1373-1393. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a8/