Strong block-block domination of a graph

Let B(G) denotes the set of all blocks of a graph G. Two blocks in G are adjacent if there is a common cutvertex incident on them. Two blocks b1, b2 ∈ B(G) are said to bb-dominate each other if there is a common vertex incident with b1 and b2. A set L ⊆ B(G) is said to be a bb-dominating set (BBD set) if every block in G is bb-dominated by some block in L. The bb-domination number γbb = γbb(G) is the cardinality of a minimum bb-dominating set of G. In this paper we define strong (weak) bb-dominating set and strong (weak) bb-full set and obtained relationship between them. We also obtain the relation with existing graph parameters.