A set \(D \subseteq E\left( G \right)\) is called edge dominating set if each not into \(D\) is adjacent with edge into \(D\) and with an edge into \(E - D\) . A Roman edge dominating function upon graph is defined to be function \(f:E \to \left\{ {0,\,1,\,2} \right\}\) fulfilling as every edge \(\left\{ {e_{u} } \right\} = 0\) is adjacent with minimum of one edge \(e_{v}\) where \(f\left( {e_{v} } \right) = 2\) . For given graph, Roman edge dominating function \(f = \left( {E_{0} ,\,E_{1} ,\,E_{2} } \right)\) is restrained Roman edge dominating function if for each edge \(e_{u}\) assuming as \(f\left( {e_{u} } \right) = 0\) is adjacent with minimum 1 edge \(e_{v}\) where \(f\left( {e_{v} } \right) = 2\) and at least one edge \(e_{w}\) for which \(f\left( {e_{w} } \right) = 0\) . Restrained Roman edge dominating function’s weight is value \(f\left( E \right) = \sum\nolimits_{{e_{u} \in E}} {f\left( {e_{u} } \right)}\) . Minimal weight of restrained Roman edge dominating function upon graph \(G\) is termed restrained Roman edge domination number of \(G\) , signified with \(\gamma_{rR}^{\prime } \left( G \right)\) . Into this paper, we obtain precise values for few standardized graphs. Additionally, we determine its connection to other graph characteristics. We conclude by proving that choice issue for limited Roman edge dominance number holds true for every generic graph, which is NP-complete.