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Semitotal domination in claw-free cubic graphs. (English) Zbl 1354.05107
Summary: In this paper, we continue the study of semitotal domination in graphs in [the authors, Discrete Math. 324, 13–18 (2014; Zbl 1284.05196)]. 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, $$\gamma_{t2}(G)$$, is the minimum cardinality of a semitotal dominating set of $$G$$. This domination parameter is squeezed between arguably the two most important domination parameters; namely, the domination number, $$\gamma (G)$$, and the total domination number, $$\gamma_{t}(G)$$. We observe that $$\gamma (G) \leq \gamma_{t2}(G) \leq \gamma_{t}(G)$$. A claw-free graph is a graph that does not contain $$K_{1,3}$$ as an induced subgraph. We prove that if $$G$$ is a connected, claw-free, cubic graph of order $$n \geq 10$$, then $$\gamma_{t2}(G) \leq 4n/11$$.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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