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-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ 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$omega(f)=sum_{vin V(G)}f(v)$. The signed total Italian k-domination number $gamma_{stI}^k(G)$ of G is the minimum weight of an STIkDF on G. In this paper we initiate the study of the signed total Italian k-dominationnumber of graphs, and we present different bounds on $gamma_{stI}^k(G)$. In addition, we determine thesigned total Italian k-domination number of some classes of graphs. Some of our results are extensions ofwell-known properties of the signed total Roman $k$-domination number $gamma_{stR}^k(G)$,introduced and investigated by Volkmann [9,12].