%0 Journal Article
%T Weak signed Roman $k$-domination in graphs
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Volkmann, Lutz
%D 2021
%\ 06/01/2021
%V 6
%N 1
%P 1-15
%! Weak signed Roman $k$-domination in graphs
%K Weak signed Roman $k$-dominating function
%K weak signed Roman $k$-domination number
%K Signed Roman $k$-dominating function
%K Signed Roman $k$-domination number
%R 10.22049/cco.2020.26734.1137
%X 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 each vertex $vin V(G)$, 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 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 $k=1$.
%U http://comb-opt.azaruniv.ac.ir/article_14019_645ee7e5ec2cd0863a1934c25c94885e.pdf