Concepts Arising from Strong Efficient Domination Number. Part I

Let G=(V,E) be a simple graph. A subset S of V(G) is called a strong (weak) efficient dominating set of G if for every v∈V(G),|N_s [v]∩S|=1.(|N_w [v]∩S|=1) , where N_s [v]={u∈V(G) ∶uv ∈E(G),deg〖u ≥degv 〗 }. (N_w [v]={u∈V(G) ∶uv ∈E(G), degv ≥degu The minimum cardinality of a strong (weak) efficient dominating set of G is called the strong (weak) efficient dominating set 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. The strong efficient bondage number b_se (G) of a non empty graph G is the minimum cardinality among all sets of edges X⊆E such that γ_se (G-X)>γ_se (G). In this paper, the strong efficient bondage number of some path related graphs and some special graphs are studied.