Edge Roman Domination in Graphs

An \emph{Edge Roman dominating function} of a graph \(G = (V, E)\) is a function \(f’ : E \to \{0,1,2\}\) satisfying the condition that every edge \(x\) for which \(f'(x) = 0\) is adjacent to at least one edge \(y\) for which \(f'(y) = 2\). The \emph{weight} of an Edge Roman dominating function is the value \(f'(E) = \sum_{x\in E} f'(x)\). The minimum weight of an Edge Roman dominating function on a graph \(G\) is called the \emph{Edge Roman domination number} of \(G\). In this paper, we initiate a study of this parameter.