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Vertices contained in all or in no minimum semitotal dominating set of a tree. (English) Zbl 1329.05232
Summary: Let $$G$$ be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, $$\gamma(G)$$, and the total domination number, $$\gamma_t(G)$$. A set $$S$$ of vertices in a graph $$G$$ 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 $$G$$. We observe that $$\gamma(G) \leq \gamma_{t2}(G) \leq \gamma_t(G)$$. We characterize the set of vertices that are contained in all, or in no minimum semitotal dominating set of a tree.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
##### Keywords:
domination; semitotal domination; trees
Full Text:
##### References:
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