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 γse (G) (γwe (G)). A graph G is strong efficient if there exists a strong efficient dominating set.