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) \( \sum_{x \in N(v)} f(x) \geq 1 \) 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.