On the signed Roman edge k-domination in graphs

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simple graph. The closed neighborhood $N_G[e]$ 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 $f:E rightarrow {-1,1,2}$ satisfying the conditions that (i) for every edge $e$ of $G$, $sum _{xin N[e]} f(x)geq k$ and (ii) every edge $e$ for which $f(e)=-1$ is adjacent to at least one edge $e'$ for which $f(e')=2$. The minimum of the values $sum_{ein E}f(e)$, 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 $gamma'_{sRk}(G)$. In this paper we establish some new bounds on the signed Roman edge $k$-domination number.