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On total domination in the Cartesian product of graphs. (English) Zbl 1392.05086
Summary: Ho proved in [P. T. Ho, Util. Math. 77, 97–100 (2008; Zbl 1161.05056)] that the total domination number of the Cartesian product of any two graphs without isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Y. Lu and X. Hou [ibid. 83, 313–322 (2010; Zbl 1242.05208)] by characterizing the pairs of graphs \(G\) and \(H\) for which \(\gamma _t \left( G\square H \right) = {1 \over 2}\gamma _t \left( G \right)\gamma _t \left( H \right)\), whenever \(\gamma_t(H) = 2\). In addition, we present an infinite family of graphs \(G_n\) with \(\gamma_t(G_n) = 2n\), which asymptotically approximate equality in \(\gamma _t \left( {G_n \square H_n } \right) \geq {1 \over 2}\gamma _t \left( {G_n } \right)^2 \).

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