Unique response Roman domination in graphs

A function f:V(G)->{0,1,2} is a Roman dominating function if every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. A function f:V(G)->{0,1,2} with the ordered partition (V"0,V"1,V"2) of V(G), where V"i={v@?V(G)|f(v)=i} for i=0,1,2, is a unique response Roman function if x@?V"0 implies |N(x)@?V"2|@?1 and x@?V"1@?V"2 implies that |N(x)@?V"2|=0. A function f:V(G)->{0,1,2} is a unique response Roman dominating function if it is a unique response Roman function and a Roman dominating function. The unique response Roman domination number of G, denoted by u"R(G), is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination number of graphs and present bounds for this parameter.