A γ-set D of a graph G is a dominating set of G with minimum cardinality. A graph is said to be γ-excellent if each vertex u of G is in some γ-set of G. A graph G is said to be γ-flexible, if to each vertex u of G, there is a γ-set of G not containing u. A dominating set D of G is said to be a bridge independent dominating set of G if ⟨D⟩ contains no bridge of G. The minimum cardinality of a bridge independent dominating set of G is called a bridge independent domination number of G and is denoted by γbi (G). A graph G is said to be γbi -excellent, if each vertex u of G is contained in some γbi -set of G. In this paper we prove that (1) Every graph G is an induced sub graph of some γbi -excellent, γ -excellent, γ -flexible graph H, with γ(G) ≤ γ(H) = γbi (H) ≤ γ(G) + 1. (2) Every γ-excellent, γ-flexible graph G is γbi -excellent and further γ(G) = γbi (G), and obtain (3) A necessary and sufficient condition under which the graph G = (G 1 ⋃ G 2) + uv, where G 1 and G 2 are disjoint γ -excellent graphs and u ∈ V (G 1), v ∈ V(G 2), is γ-excellent. © 2004 Taylor & Francis Group, LLC.