Signed total Italian domination in graphs

Abstract

A signed total Italian dominating function (STIDF) of a graph $G$ with vertex set $V(G)$ is defined as a function $f:V(G)\to \{-1,1,2\}$ having the property that (i) $\sum _{x\in N(v)}f(x)\ge 1$ for each $v\in V(G)$, where $N(v)$ is the neighborhood of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ is adjacent to a vertex $v$ for which $f(v)=2$ or adjacent to two vertices $w$ and $z$ with $f(w)=f(z)=1$. The weight of an STIDF is the sum of its function values over all vertices. The \textit{signed total Italian domination number} of $G$, denoted by ${\gamma }_{stI}(G)$, is the minimum weight of an STIDF in $G$. We initiate the study of the signed total Italian domination number, and we present different sharp bounds on ${\gamma }_{stI}(G)$. In addition, we determine the signed total Italian domination number of some classes of graphs.