On the signed Roman edge $k$-domination in graphs

Abstract

Let $k\geq 1$ 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 $G$‎, ‎$\sum _{x\in N[e]} f(x)\geq k$ 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 $\sum_{e\in E}f(e)$‎, ‎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.

Keywords

• Signed Roman edge $k$-dominating function‎
• ‎Signed Roman edge $k$-domination number