a. p. kazemi
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For any integer $kgeq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$, we define a function $f:Vrightarrow {0,1,2}$ as a Roman $k$-tuple dominating function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least $k$ and for any vertex $v$ with $f(v)neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$. The minimum weight of a Roman $k$-tuple dominating function $f$ on $G$ is called the Roman $k$-tuple domination number of the graph where the weight of $f$ is $f(V)=sum_{vin V}f(v)$. In this paper, we initiate to study the Roman $k$-tuple domination number of a graph, by giving some sharp bounds for the Roman $k$-tuple domination number of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding the Roman $k$-tuple domination number of some known graphs is our other goal. Some of our results extend these one given by Cockayne and et al. cite{CDHH04} in 2004 for the Roman domination number.
Keywords: Roman $k$-tuple domination number, Roman $k$-tuple graph, $k$-Tuple domination number, $k$-Tuple total domination number, Mycieleskian of a graph -
For every positive integer k, a set S of vertices in a graph G = (V, E) is a k-tuple dominating set of G if every vertex of V − S is adjacent to at least k vertices and every vertex of S is adjacent to at least k−1 vertices in S. The minimum cardinality of a k-tuple dominating set of G is the k-tuple domination number of G. When k = 1, a k-tuple domination number is the well-studied domination number. We define the k-tuple domatic number of G as the largest number of sets in a partition of V into k-tuple dominating sets. Recall that when k = 1, a k-tuple domatic number is the well-studied domatic number. In this study, basic properties and bounds for the k-tuple domatic number are derived.
Keywords: k-tuple dominating set, k-tuple domination number, ktuple domatic number
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