Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601Weak signed Roman $k$-domination in graphs1151401910.22049/cco.2020.26734.1137ENLutz VolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20191207Let $k\ge 1$ 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, LĂ¶wenstein and Samodivkin [1] for the case $k=1$.https://comb-opt.azaruniv.ac.ir/article_14019_645ee7e5ec2cd0863a1934c25c94885e.pdf