Strong efficient domination and strong independent saturation number of graphs

A subset S of V(G) of a graph G is called a strong (weak) efficient dominating set of G if for every v V(G), │Ns[v]∩S│= 1 (│Nw[v]∩S│= 1 ) where Ns(v) = { u V(G) : uv E(G), deg(u) ≥ deg(v) } and Nw(v) = { u V(G) : uv E(G), deg(v) ≥ deg(u)}, Ns[v] = Ns(v) {v}, Nw[v] = Nw(v) {v}. The minimum cardinality of a strong (weak) efficient dominating set is called strong (weak) efficient domination number of G and is denoted by se (G) ( we (G)). A graph G is strong efficient if there exists a strong efficient dominating set.