In this paper, the concept of Minimal Roman Dominating Function is considered. A characterization of a Minimal Roman Dominating Function has been given. It has also been proved that for any graph with vertices, the Upper Roman Domination Number is . It is also shown that if is a Minimal Roman Dominating Function then 2 − is a Roman Dominating Function for a graph without isolated vertices. We also prove that Υ ( ) = 2 Υ( ) if and only if there is a Minimum Roman Dominating Function which does not assume value 1 at any vertex of .