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A set S of vertices in G 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 , γ t 2 ( G ), is the minimum cardinality of a semitotal dominating set of G . Clearly, γ ( G ) ≤ γ t 2 ( G ) ≤ γ t ( G ). In this paper, for any nontrivial tree T that is not a star, we investigate the ratios γ t 2 ( T )/ γ ( T ) and γ t ( T )/ γ t 2 ( T ), and provide constructive characterizations of trees achieving the upper bounds.
Wei Zhuang (2024) studied this question.