# Weak signed Roman $k$-domination in graphs

Document Type : Original paper

Author

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 weak signed Roman $k$-dominating function (WSRkDF) 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 closed neighborhood of $v$. The weight of a WSRkDF $f$ is $w(f)=\sum_{v\in V(G)}f(v)$. The weak signed Roman $k$-domination number $\gamma_{wsR}^k(G)$ of $G$ is the minimum weight of a WSRkDF on $G$. In this paper we initiate the study of the weak signed Roman $k$-domination number of graphs, and we present different bounds on $\gamma_{wsR}^k(G)$. In addition, we determine the weak signed Roman $k$-domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed Roman $k$-domination number $\gamma_{sR}^k(G)$, introduced and investigated by Henning and Volkmann [5] as well as Ahangar, Henning, Zhao, Löwenstein and Samodivkin [1] for the case $k=1$.

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#### References

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