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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.)
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