Abstract: In this paper, we introduce a new concept of edge domination graph named while relatively prime edge dominating set of a graph G. Let G (V, E) be a graph. For a set D′ ⊆E(G) is said to be relatively prime edge dominating set, if it is an edge dominating set with some two elements and for each pair of edge (e, f) ∈ D′(????) such that (deg e, deg f) = 1 (where degree of edge d(e) = deg u + deg v-2, e(=uv) in E(G)) . The minimum cardinality of a relatively prime edge dominating set (rped-set) is called relatively prime domination number and it is denoted by ???????????????????? ′ (G). In this paper, we compute about an edge dominating graph of relatively prime.