Strong (weak) VV-dominating set of a graph

Two vertices u, w ∈ V vv-dominate each other if they incident on the same block. A vertex u ∈ V strongly vv-dominates a vertex w ∈ V if u and w, vv-dominate each other and d_vv(u) ≥ d_vv(w). A set of vertices is said to be strong vv-dominating set if each vertex outside the set is strongly vv-dominated by at least one vertex inside the set. The strong vv-domination number γ_svv(G) is the order of the minimum strong vv-dominating set of G. Similarly weak vv-domination number γ_wvv(G) is defined. We investigate some relationship between these parameters and obtain Gallai’s theorem type results. Several upper and lower bounds are established. In addition, we characterize the graphs attaining some of these bounds.