Signed Roman -Domination in Graphs

Let k >= 1 be an integer, and let G be a finite and simple graph with vertex set V(G). A signed Roman k-dominating function (SRkDF) on a graph G is a function f : V(G). {-1, 1, 2} satisfying the conditions that (i) Sigma(x is an element of N[v]) f(x) >= k for each vertex v is an element of V(G), where N[v] is the closed neighborhood of v, and (ii) every vertex u for which f (u) = -1 is adjacent to at least one vertex v for which f (v) = 2. The weight of an SRkDF f is w(f) = Sigma(v is an element of V(G)) f (v). The signed Roman k-domination number gamma(k)(SR) (G) of G is the minimum weight of an SRkDF on G. In this paper we initiate the study of the signed Roman k-domination number of graphs, and we present different bounds on gamma(k)(SR)(G). In addition, we determine the signed Roman k-domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed Roman domination number gamma(sR)(G) = gamma(1)(SR) (G), introduced and investigated by Ahangar et al. (J Comb Optim doi: 10.1007/s10878-012-9500-0, 2014).