Weak signed Roman k-domination in graphs

Let $kge 1$ 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 $sum_{xin N[v]}f(x)ge k$ for eachvertex $vin V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is$w(f)=sum_{vin V(G)}f(v)$. The weak signed Roman $k$-domination number $gamma_{wsR}^k(G)$ of $G$ is theminimum weight of a WSRkDF on $G$. In this paper we initiate the study of the weak signed Roman $k$-dominationnumber of graphs, and we present different bounds on $gamma_{wsR}^k(G)$. In addition, we determine theweak signed Roman $k$-domination number of some classes of graphs. Some of our results are extensions ofwell-known properties of the signed Roman $k$-domination number $gamma_{sR}^k(G)$,introduced and investigated by Henning and Volkmann cite{hv16} as well as Ahangar, Henning, Zhao, L"{o}wenstein andSamodivkin cite{ahzls} for the case $k=1$.