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 dvv(u) ≥ dvv(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 γsvv(G) is the order of the minimum strong vv-dominating set of G. Similarly weak vv-domination number γwvv(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.
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