signed total italian k-domination in graphs

Let k≥1 be an integer, and let g be a finite and simple graph with vertex set v(g). a signed total italian k- ominating function (stikdf) on a graph g is a function f:v(g)→{−1,1,2} satisfying the conditions that ∑x∈n(v)f (x)≥k for each vertex v∈v(g), where n(v) is the neighborhood of v, and each vertex u with f(u)=−1 is adjacent to a vertex v with f(v)=2 or to two vertices w and z with f(w)=f(z)=1. the weight of an stikdf f is ω (f)=∑v∈v(g)f(v). the signed total italian k-domination number γksti(g) of g is the minimum weight of an stikdf on g. in this paper we initiate the study of the signed total italian k-domination number of graphs, and we present different bounds on γksti(g). in addition, we determine the signed total italian k-domination number of some classes of graphs. some of our results are extensions of well-known properties of the signed total roman k - domination number γkstr(g), introduced and investigated by volkmann [9,12].