Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Articles in Press 2025 06 13 Signed total Italian $k$-domination 14972 10.22049/cco.2025.30392.2449 EN Qin CHEN College of Science, China Jiliang University, Hangzhou 310018, China Journal Article 2025 03 02 Let $k\geq 1$ be an integer. A signed total Italian $k$-dominating function (STIkDF) on a graph $G=(V, E)$ is a function $f: V\rightarrow \{-1,1,2\}$ satisfying the conditions that $\sum_{u\in N(v)}f(u)\geq k$ for each vertex $v\in V$, 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 $w(f)=\sum_{v\in V}f(v)$. The signed total Italian $k$-domination number of $G$, denoted by $\gamma_{stI}^k(G)$, is the minimum weight of an STIkDF on $G$. In this paper, we prove that the decision problem for the signed total $k$-domination is NP-complete for $k\in\{1,2\}$. We present tight lower bound on \textcolor{red}{$\gamma_{stI}^2(G)$}, and characterize all extremal graphs. Using a discharging method, we also determine the value \textcolor{red}{$\gamma_{stI}^2(C_3\Box C_n)$} for all $n\geq 3$. https://comb-opt.azaruniv.ac.ir/article_14972_d5751b80e02d2119e4cd9f5cfb6a6425.pdf