Signed Roman edge domination numbers in graphs

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