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On matching and semitotal domination in graphs. (English) Zbl 1284.05196
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 $$G$$ 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 $$\gamma (G)\leq \gamma_{t2}(G)\leq\gamma_ t(G)$$. It is known that $$\gamma (G)\leq\alpha '(G)$$, where $$\alpha '(G)$$ denotes the matching number of $$G$$. However, the total domination number and the matching number of a graph are generally incomparable.
We provide a characterization of minimal semitotal dominating sets in graphs. Using this characterization, we prove that if $$G$$ is a connected graph on at least two vertices, then $$\gamma_{t2}(G)\leq\alpha '(G)+1$$ and we characterize the graphs achieving equality in the bound.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
domination; total domination; semitotal domination; matching
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##### References:
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