Let G = (V, E) be a simple graph with p vertices and q edges. A subset S of V (G) is called a strong (weak) efficient dominating set of G if for every v ∈ V (G) we have |N_s[v] ∩ S|= 1 (resp. |N_w [v] ∩ S|= 1), where N_s(v) = {u ∈ V (G) : uv ∈ E(G), deg(u) ≥ deg(v)}. The minimum cardinality of a strong (weak) efficient dominating set of G is called the strong (weak) efficient domination number of G and is denoted by γ_se(G) (γ_we(G)). A graph G is strong efficient if there exists a strong efficient dominating set of G. In this paper, some cycle and star related Nordhaus-Gaddum type relations on strong efficient dominating sets and the number of strong efficient dominating sets are studied.

Keywords: strong efficient dominating sets, strong efficient domination number and number of strong efficient dominating sets.