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Two vertices  vv-dominate each other if they incident on the same block. A vertex  strongly vv-dominates a vertex  if u and w, vv-dominate each other and  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 is the order of the minimum strong vv-dominating set of G. Similarly weak vv-domination number  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.

THE STRONG (WEAK) VV-DOMINATING SET OF A GRAPH