Excellent Domination in Fuzzy Graphs

Let G be a fuzzy graph. A subset D of V is said to be Fuzzy dominating set if every vertex u ∈ V (G) there exists a vertex v ∈ V −D such that uv ∈ E(G) and μ(uv) 6 σ(u) ∧ σ(v). The minimum Cardinality of fuzzy dominating set is denoted by γf . A graph G is said to be fuzzy excellent if every vertex of G belongs to γf -sets of G. In this paper, we give a construction to imbedded non-excellent fuzzy graph G in an excellent fuzzy graph H such that γf (H) 6 γf (G) + 2. We also show that for a given non-excellent fuzzy graph G, there is subdivision of G which is fuzzy excellent. Also, we introduce the concept of γf -flexible, fuzzy bridge independent dominating number γ bi and obtain some interesting results for this new parameter in excellent fuzzy graphs.