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Total $\{k\}$-domination in special graphs

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  • For a positive integer $k$ and a graph $G = (V,E)$, a function $f:V \to \{0,1,...,k\}$ is called a total $\{k\}$-dominating function of $G$ if $\sum_{u∈ N_G(v)}f(u)≥ k$ for each $v∈ V$, where $N_G(v)$ is the neighborhood of $v$ in $G$. The total $\{k\}$-domination number of $G$, denoted by $\gamma _t^{\left\{ k \right\}}\left( G \right)$, is the minimum weight of a total $\{k\}$-dominating function $G$, where the weight of $f$ is $\sum_{v∈ V}f(v)$. In this paper, we determine the exact values of the total $\{k\}$-domination number for several commonly-encountered classes of graphs including cycles, paths, wheels, and pans.

    Mathematics Subject Classification: 05C10.

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