Some remarks on the signed total Italian $k$-domination number of graphs

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Document Type : Original paper

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RWTH Aachen University, 52056 Aachen, Germany

Abstract

Let $k\ge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$. Volkmann \cite{vo21} defined the signed total Italian $k$-dominating function (STIkDF) on a graph $G$ as a function $f:V(G)\rightarrow\{-1,1,2\}$ satisfying the conditions that $\sum_{x\in N(v)}f(x)\ge k$ for each vertex $v\in V(G)$, where $N(v)$ is the neighborhood of $v$, and every vertex $u$ for which $f(u)=-1$ is adjacent to at least one 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 STIkDF $f$ is $w(f)=\sum_{v\in V(G)}f(v)$. 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 continue the study of the signed total Italian $k$-domination number. We present new bounds on $\gamma_{stI}^k(G)$, and we determine the signed total Italian $k$-domination number of some complete $p$-partite graphs. Furthermore, we show that the difference $\gamma_{stR}^k(G)-\gamma_{stI}^k(G)$ can be arbitrarily large, where $\gamma_{stR}^k(G)$ is the signed total Roman $k$-domination number.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 01 September 2025

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