Signed Roman Edge k -Domination in Graphs

Let $k \ge 1$ 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 $G$, $\Sigma_{ x \in N_G [e] } f(x) \ge k$ 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 $\Sigma_{e \in E} f(e)$, 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 initiate the study of the signed Roman edge $k$-domination in graphs and present some (sharp) bounds for this parameter

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, ∑x∈NG[e]f(x) ≥ 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 ∑e∈Ef(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 initiate the study of the signed Roman edge k-domination in graphs and present some (sharp) bounds for this parameter

On the signed Roman edge kk-domination in graphs

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