On the signed Roman k-domination in graphs

Let k ≥ 1 be an integer and G be a simple and finite 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} such that (i) every vertex v with f(v) = −1 is adjacent to at least one vertex u with f(u) = 2, (ii) ∑ u∈N[v] f(u) ≥ k holds for any vertex v. The weight of a SRkDF f is ∑ u∈V (G) f(u), and the minimum weight of a SRkDF is the signed Roman k-domination number γ k sR(G) of G. In this paper, we investigate the signed Roman k-domination number of graphs, and we establish some bounds on γ k sR(G). In the case that T is a tree, we present lower and upper bounds on γ k sR(T) for k ∈ {3, 4} and classify all extremal trees. Mathematics Subject Classification (2010): 05C69. Ke words: Signed Roman k-dominating function, signed Roman k-domination numbe