M . Atapour , S . M . Sheikholeslami and L . Volkmann SIGNED STAR { k } - DOMATIC NUMBER OF A GRAPH

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {±1,±2, . . . ,±k} is said to be a signed star {k}-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. The signed star {k}-domination number of a graph G is γ{k}SS(G) = min{ ∑ e∈E f(e) | f is a SS{k}DF on G}. A set {f1, f2, . . . , fd} of distinct signed star {k}-dominating functions on G with the property that ∑d i=1 fi(e) ≤ k for each e ∈ E(G), is called a signed star {k}-dominating family (of functions) on G. The maximum number of functions in a signed star {k}-dominating family on G is the signed star {k}-domatic number of G, denoted by d{k}SS(G). In this paper we study the properties of the signed star {k}domination number γ{k}SS(G) and signed star {k}-domatic number d{k}SS(G). In particular, we determine the signed star {k}domination number of some classes of graphs. Some of our results extend these one given by Xu [7] for the signed star domination number and Atapour et al. [1] for the signed star domatic number.