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$$k$$-tuple total domination in cross products of graphs. (English) Zbl 1261.90073
Summary: For $$k\geq 1$$ an integer, a set $$S$$ of vertices in a graph $$G$$ with minimum degree at least $$k$$ is a $$k$$-tuple total dominating set of $$G$$ if every vertex of $$G$$ is adjacent to at least $$k$$ vertices in $$S$$. The minimum cardinality of a $$k$$-tuple total dominating set of $$G$$ is the $$k$$-tuple total domination number of $$G$$. When $$k=1$$, the $$k$$-tuple total domination number is the well-studied total domination number. In this paper, we establish upper and lower bounds on the $$k$$-tuple total domination number of the cross product graph $$G\times H$$ for any two graphs $$G$$ and $$H$$ with minimum degree at least $$k$$. In particular, we determine the exact value of the $$k$$-tuple total domination number of the cross product of two complete graphs.

##### MSC:
 90C35 Programming involving graphs or networks
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##### References:
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