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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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