Signed total Italian k-domination in graphs

Document Type : Original paper

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RWTH Aachen University

Abstract

Let $k\ge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$. A signed total Italian $k$-ominating function (STIkDF) on a graph $G$ is 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 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 $\omega(f)=\sum_{v\in V(G)}f(v)$. 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 initiate the study of the signed total Italian $k$-domination number of graphs, and we  present different bounds on $\gamma_{stI}^k(G)$. In addition, we determine the
signed total Italian $k$-domination number of some classes of graphs.  Some of our results are extensions of well-known properties of the signed total Roman $k$-domination number $\gamma_{stR}^k(G)$, introduced and investigated by Volkmann [9,12].

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References

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Communications in Combinatorics and Optimization
Volume 6, Issue 2
December 2021
Pages 171-183

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