Geodesic
Semitotal domination in claw-free graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1585-1605

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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 γ_t2(G), is the minimum cardinality of a semitotal dominating set in G. We prove that if G is a connected claw-free graph of order n with minimum degree δ(G)≥ 2 and is not one of fourteen exceptional graphs (ten of which are cycles), then γ_t2(G) ≤37n, and we also characterize the graphs achieving equality, which are an infinite family of graphs. In particular, if we restrict δ(G) ≥ 3 and G K_4, then we can improve the result to γ_t2(G) ≤25n, solving the conjecture for the case of claw-free graphs, proposed by Goddard, Henning and McPillan in [Semitotal domination in graphs, Util. Math. 94 (2014) 67–81].
Keywords: semitotal domination, minimum degree, claw-free graphs 
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Chen, Jie; Liang, Yi-Ping; Xu, Shou-Jun. Semitotal domination in claw-free graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1585-1605. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a18/