On the signed Roman edge k-domination in graphs

Let k ≥ 1 be an integer, and G = (V, E) be a finite and simple graph. The closed neighborhood NG[e] of an edge e in a graph G is the set consisting of e and all edges having a common end-vertex with e. A signed Roman edge k-dominating function (SREkDF) on a graph G is a function f : E → {−1, 1, 2} satisfying the conditions that (i) for every edge e of G, Σ x∈N[e] f(x) ≥ k and (ii) every edge e for which f(e) = −1 is adjacent to at least one edge e 0 for which f(e′) = 2. The minimum of the values Σ e∈E f(e), taken over all signed Roman edge k-dominating functions f of G, is called the signed Roman edge k-domination number of G and is denoted by γ′sRk(G). In this paper we establish some new bounds on the signed Roman edge k-domination number.