Geodesic
On the total restrained domination number of direct products of graphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 629-641

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Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V∖S is adjacent to a vertex in S as well as to another vertex in V∖S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γ_r^t(G), is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds are sharp by presenting some infinite families of graphs that attain these bounds.
Keywords: total domination number, total restrained domination number, direct product of graphs 
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Shiu, Wai; Chen, Hong-Yu; Chen, Xue-Gang; Sun, Pak. On the total restrained domination number of direct products of graphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 629-641. http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a2/