© 2019, IJCSE All Rights Reserved 21 International Journal of Scientific Research in ______________________________ Research Paper . Mathematical and Statistical Sciences Vol.6, Issue.2, pp.21-25, April (2019) E-ISSN: 2348-4519 DOI: https://doi.org/10.26438/ijsrmss/v6i2.2125 Gallai-Type Theorems in Gallai Fuzzy Graphs on Domination Parameters M. Kaliraja 1* , G. Karlmarx 2 1 Dept. of Mathematics, H.H. The Rajah's college (autonomous), Pudukkottai, Tamil Nadu, India 2 Dept. of Mathematics, Syed Ammal Arts and Science College, Ramanathapuram, Tamil Nadu, India * Corresponding Author: mkr.maths009@gmail.com Available online at: www.isroset.org Received: 03/Mar/2019, Accepted: 14/Apr/2019, Online: 30/Apr/2019 Abstract— The Gallai fuzzy graph Γ(G) of a fuzzy graph G has the fuzzy edges of G as its fuzzy vertices and two distinct fuzzy edges of G are fuzzy incident in G, but do not span a fuzzy triangle in G. The Gallai fuzzy graphs are fuzzy spanning Gallai sub graphs of the well-known Class of fuzzy line graphs. Let γ(Γ(G)) and i(Γ(G)) denote the minimum fuzzy cardinality of a fuzzy dominating set of a Gallai fuzzy graph Γ(G)= (σ, μ) with n fuzzy vertices a nd maximum fuzzy degree Δ(Γ(G)), (Γ(G))≤ n- Δ(Γ(G), i(G) ≤ n-Δ(Γ(G)). In this manuscript, we characterized the fuzzy connected bipartite Gallai fuzzy graphs which achieve this upper bound. Here, we have shown that an arbitrary Gallai fuzzy Graph G are furnished with two conditions which are necessary if γ(Γ(G)) + Δ(Γ(G))= n and are sufficient to achieve n - 1 ≤ γ(Γ(G))+Δ(Γ(G)) ≤ n. Keywords— Gallai fuzzy graph, Gallai-type theorems, fuzzy domination parameters I. INTRODUCTION The study of dominating sets in graphs was started by two different authors, viz. Ore, 1962 and Berge, 1962. In 1965, L.A. Zadeh [1] introduced a mathematical frame work to explain the concepts of uncertainty in real life through the publication of a seminal paper. In 1975, A. Rosenfeld [2] introduced the notation of fuzzy graph theoretic concept such as paths, cycles and connectedness. In 1977, Cockayne and Hedetniemi have introduced the domination number and independent domination. In 1996, Van Bang Le [3] discussed about the Gallai graphs and anti-Gallai graphs. Similarly, S. Aparna Lakshmanan and S.B. Rao [4] also deliberated the Gallai graphs and anti-Gallai graphs. Further, A. Somasundram and S. Somasundram [5] have explored the domination in fuzzy graphs. In addition, the domination, independent and irredundance numbers were discussed by A. Nagoorgani and P. Vadivel [6]. In this paper, we have analyzed about the Fuzzy domination in Gallai fuzzy graph based on the concept of Gallai type theorems and domination parameters by Gayla S. Domke et.al., [7] and some important Gallai-type theorems on fuzzy domination. However, the Gallai-type theorems involving lower fuzzy domination parameters combined with maximum fuzzy degree and the relationship with other parameters have also been established in this manuscript. II. PRELIMINARIES Definition 2.1 Let G={σ, μ} be a fuzzy graph, H ={ σ , μ  } is a fuzzy sub graph of G if σ  is a suset of σ and μ  is a subset μ; that is if σ′(x) ≤ σ(x) for every x ϵv and μ′(e) ≤ μ(e) for every e ϵE. (σ′, μ′) is a fuzzy spanning sub graph of (σ, μ) if σ′ = σ and μ′ is a subset of μ: that is σ′(x) = σ(x) for every xϵV and μ′(e) ≤ μ(e) for every e ϵ E. For any fuzzy subset X of V such that X is the fuzzy sub graph of σ , the fuzzy sub graph of (σ, μ) induced fuzzy graph by X is the maximal fuzzy sub graph of (σ, μ), that has fuzzy vertex set X and it is the fuzzy sub graph (X,Ʈ ), where Ʈ(X, Y ) = X(x) ˄ X(y) ˄ μ(x, y) for all x, y in V. Definition 2.2 The complement of a fuzzy graph G denoted by is defined to be = (σ, ), where = σ(x) ^ σ(y) -μ(x, y). Definition 2.3 Let G = (σ, μ) be a fuzzy graph. The degree of a fuzzy Vertex x is defined as d(x) =Σ x y, xϵv μ(x, y), If is also denoted as d(x). A fuzzy graph is said to be fuzzy regular if every fuzzy vertex is of same degree. Since μ(x, y) > 0 for xyϵ E and μ(x, y) = 0 for xy is does not belongs to E, this is equivalent to d(x) =Σ xy ϵ E μ(x y), the minimum and maximum degree of G are δ (G)= ^{ d(x), x ϵ v } and Δ (G)=˅ {d(x), x ϵ v}. Example 2.4 A fuzzy graph with G as the underlying set is a finite non- empty unordered pair of G = (σ, μ), where σ: V → [0, 1] is a fuzzy subset, μ: → [0, 1] is a fuzzy relation on the fuzzy subset σ such that μ(x, y) ≤ σ(x) ˄ σ(y) for all x, y  V where ˄ and ˅ stands for minimum and maximum.The underlying