Strong (weak) domination and independent strong (weak) domination in bipolar fuzzy graphs

Let u and v be any two vertices of bipolar fuzzy graphs G. Then u strongly dominates v (v weakly dominates u) if (i) effective edge between u and v and (ii) dN (u) ≥ dN (v). A subset S of V is called a strong (weak) dominating set in bipolar fuzzy graphs G for every v εV-S there exist u ε S such that u strongly dominates v (v weakly dominates u) and it is denoted by SBFD-set (WBFD-set). A strong (weak) dominating set S of bipolar fuzzy graphs G is said to be minimal strong (weak) dominating set if no proper subset of S is a strong (weak) dominating set of The minimum cardinality among all minimal strong (weak) dominating set is called strong (weak) bipolar fuzzy domination number of G, and is denoted by γ^s(γ^w). A SBFD-set (WBFD-set) S of bipolar fuzzy graphs G is said to be an independent strong (weak) dominating set of G, if it is independent. The minimum cardinality of an independent strong (weak) dominating set is called the independent strong (weak) bipolar fuzzy dominating number and it is denoted by γ^is(γ^iw). In this paper, the strong (weak) domination number and independent strong (weak) domination number in bipolar fuzzy graphs are introduced and its bounds are obtained. Also the strong (weak) dominating set and independent strong (weak) dominating set are characterised.