RELATIONSHIP between BLOCK DOMINATION PARAMETERS of A GRAPH

Two vertices u,w ϵ 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 = γ_vv(G) is the cardinality of a minimum VVD-set of G. Two blocks b1, b2 ϵ 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 γ_bb = γ_bb(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 γ_vb = γ_vb(G) (bv-domination number γ_bv = γ_bv(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.