Signed Roman edge domination numbers in graphs

Abstract

The closed neighborhood $N_G[e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and of all edges having a common end-vertex with $e$. Let $f$ be a function on $E(G)$, the edge set of $G$, into the set $\{-1, 1, 2\}$. If $\sum _{x\in N[e]}f(x) \ge 1$ for every edge $e$ of $G$ and every edge $e$ for which $f (e) = -1$ is adjacent to at least one edge $e'$ for which $f (e')= 2$, then $f$ is called a signed Roman edge dominating function of $G$. The minimum of the values $\sum _{e\in E(G)} f(e)$, taken over all signed Roman edge dominating functions $f$ of $G$, is called the signed Roman edge domination number of $G$ and is denoted by $\gamma _{sR}'(G)$. In this note we initiate the study of the signed Roman edge domination in graphs and present some (sharp) bounds for this parameter.