Signed total Italian $k$-domination in digraphs

Document Type : Original paper

Author

Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

Abstract

Let $k\ge 1$ be an integer, and let $D$ be a finite and simple digraph with vertex set $V(D)$. A signed total Italian $k$-dominating function (STIkDF) on a digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-(v)}f(x)\ge k$ for each vertex $v\in V(D)$, where $N^-(v)$ consists of all vertices of $D$ from which arcs go into $v$, and (ii) each vertex $u$ with $f(u)=-1$ has an in-neighbor $v$ for which $f(v)=2$ or two in-neighbors $w$ and $z$ with $f(w)=f(z)=1$. The weight of an STIkDF $f$ is $\omega(f)=\sum_{v\in V(D)}f(v)$. The signed total Italian $k$-domination number $\gamma_{stI}^k(D)$ of $D$ is the minimum weight of an STIkDF on $D$. In this paper we initiate the study of the signed total Italian $k$-domination number of digraphs, and we  present different bounds on $\gamma_{stI}^k(D)$. In addition, we determine the signed total Italian $k$-domination number of some classes of digraphs.

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References

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Communications in Combinatorics and Optimization
Volume 9, Issue 2
June 2024
Pages 339-351

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