weak signed roman k-domination in graphs

Let k≥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)→{−1,1,2} satisfying the conditions that ∑x∈n[v]f(x)≥k for each vertex v∈v(g), where n[v] is the closed neighborhood of v. the weight of a wsrkdf f is w(f)=∑v∈v(g)f(v). the weak signed roman k-domination number γ^k wsr(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 γ^k wsr(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 γksr(g), introduced and investigated by henning and volkmann [5] as well as ahangar, henning, zhao, löwenstein and samodivkin [1] for the case k=1.