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A Vizing-type result for semi-total domination. (English) Zbl 1407.05172
Summary: A set of vertices $$S$$ in a simple isolate-free graph $$G$$ is a semi-total dominating set of $$G$$ if it is a dominating set of $$G$$ and every vertex of $$S$$ is within distance 2 of another vertex of $$S$$. The semi-total domination number of $$G$$, denoted by $$\gamma_{t 2}(G)$$, is the minimum cardinality of a semi-total dominating set of $$G$$. In this paper, we study semi-total domination of Cartesian products of graphs. Our main result establishes that for any graphs $$G$$ and $$H$$, $$\gamma_{t 2}(G \square H) \geq \frac{1}{3} \gamma_{t 2}(G) \gamma_{t 2}(H)$$.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C76 Graph operations (line graphs, products, etc.)
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