Abstract: A dominating set of a graph is roman dominating set, if and are two subsets of and satisfying the condition that every vertex in is adjacent to exactly one to one a vertex in as well as adjacent to some vertex in . The roman domination number of of is the minimum cardinality of a roman dominating set of . In this paper many bounds on are obtained and its exact values for some standard graphs are found. Also relationship with other parameter is investigated. We introduce Roman dominating set in which the interest is in dominating contains two types of subset.