On roman domination number of functigraph and its complement

Abstract Let be a graph and be a function where for every vertex with there is a vertex where Then is a Roman dominating function or a of The weight of is The minimum weight of all is called the Roman domination number of denoted by Let be a graph with and G' be a copy of with Then a functigraph with function is denoted by its vertices and edges are and respectively. This paper deals with the Roman domination number of the functigraph and its complement. We present a general bound where is a permutation. Also, the Roman domination number of some special graphs are considered. We obtain a general bound of and we show that this bound is sharp.