Two vertices u, w ∈ V, vv-dominate each other if they are incident on the same block. A set S ⊆ V is a vv-dominating set (VVD-set) if every vertex in V – S is vv-dominated by a vertex in S. The vv-domination number γvv = γvv(G) is the cardinality of a minimum VVD-set of G. Two blocks b1, b2 ∈ B(G) the set of all blocks of G, bb-dominate each other if there is a common cutpoint. A set L ⊆ B(G) is said to be a bb-dominating set (BBD set) if every block in B(G) – L is bb-dominated by some block in L. The bb-domination number γbb = γbb(G) is the cardinality of a minimum BBD-set of G. A vertex v and a block b are said to b-dominate each other if v is incident on the block b. Then vb-domination number γvb = γvb(G) (bv-domination number γbv = γbv(G)) is the minimum number of vertices (blocks) needed to b-dominate all the blocks (vertices) of G. In this paper we study the properties of these block domination parameters and establish a relation between these parameters giving an inequality chain consisting of nine parameters.

The concept of inverse domination was introduced by Kulli V.R. and Sigarakanti S.C. [9] . Let D be a  -set of G. A dominating set D 1  V-D is called an inverse dominating set of G with respect to D. The inverse domination number   (G) is the order of a smallest inverse dominating set. Motivated by this definition we define another parameter as follows. Let D be a maximum independent set in G. An independent set S  V-D is called an inverse independent set with respect to D. The inverse independence Number β 0 -1 (G) = max { S : S is an inverse independent set of G}.We find few bounds on inverse domination number and also initiate the study of the inverse independence number giving few bounds on inverse independence number of a graph.