ON THE TOTAL {k}-DOMINATION AND TOTAL {k}-DOMATIC NUMBER OF GRAPHS
نویسنده
چکیده
For a positive integer k, a total {k}-dominating function of a graph G without isolated vertices is a function f from the vertex set V (G) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (G), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) is the open neighborhood of v. The weight of a total {k}-dominating function f is the value ω(f) = ∑ v∈V f(v). The total {k}-domination number, denoted by γ {k} t (G), is the minimum weight of a total {k}-dominating function on G. A set {f1, f2, . . . , fd} of total {k}-dominating functions on G with the property that ∑d i=1 fi(v) ≤ k for each v ∈ V (G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by d {k} t (G). Note that d {1} t (G) is the classic total domatic number dt(G). In this paper, we present bounds for the total {k}-domination number and total {k}domatic number. In addition, we determine the total {k}-domatic number of cylinders and we give a Nordhaus-Gaddum type result.
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