Ratio Mathematica Volume 47, 2023 Connected end anti-fuzzy equitable dominating set in anti-fuzzy graphs S. Firthous Fatima* K. Janofer† Abstract In this paper, the notion of connected end anti-fuzzy equitable dominating set of an anti- fuzzy graph is discussed. The connected end anti-fuzzy equitable domination number for some standard graphs are obtained. The relation between anti-fuzzy equitable domination number, end anti-fuzzy equitable domination number and connected end anti-fuzzy equitable domination number are established. Theorems related to these parameters are stated and proved. Keywords: dominating set; end anti-fuzzy equitable dominating set; connected end anti-fuzzy equitable dominating set 2010 AMS subject classification: 05C62, 05E99, 05C07‡ * Assistant Professor, Department of Mathematics, Sadakathullah Appa College (Autonomous), Rahmath Nagar, Tirunelveli, India, 627011; kitherali@yahoo.co.in. † Part-Time Research Scholar, Reg. No: 19131192092019, Department of Mathematics, Sadakathullah Appa College (Autonomous), Rahmath Nagar, Tirunelveli, India, 627011, Affiliated to Manonmaniam Sundaranar University, Abishekaptti, Tirunelveli 627012, India; janofermath@gmail.com. ‡Received on September 25, 2022. Accepted on March 1, 2023. Published on March 21, 2023. DOI:10.23755/rm.v39i0.861. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 271 mailto:kitherali@yahoo.co.in mailto:janofermath@gmail.com S. Firthous Fatima and K. Janofer 1. Introduction A graph is an advantageous method for representing the data including connection between objects. The objects are represented by nodes and relations by arcs. Whenever there is uncertainty or vagueness in the description of items or in its connections or in both, it is common that we have to plan an anti-fuzzy graph model. A fuzzy set, as a superset of a crisp set, owes its origin to the work of Zadeh [14] in 1965 that has been introduced to deal with uncertainty. M. Akram [1] defined the concept of anti-fuzzy graph structures in 2012. A. Somasundaram and S.Somasundaram [12] presented several types of domination parameters such as independent domination, total domination, connected domination and domination in Cartesian product and composition of fuzzy graphs. R.Muthuraj and A. Sasireka [7] introduced domination in anti-fuzzy graphs. The concept of equitable domination in graphs was introduced by Swaminathan and Dharmalingam [13]. The end equitable domination number in graph has introduced by J.H.Hattingh and M.H.Henning [8]. Further results were extended by Murthy and Puttaswamy [10]. Some works in extension of fuzzy graphs can be found in [5, 6, 9]. S.Firthous Fatima and K.Janofer [2, 3, 4] introduced the concept of anti-fuzzy equitable dominating set, connected anti-fuzzy equitable dominating set and end anti- fuzzy equitable dominating set of an anti-fuzzy graphs. In this paper, the connected end anti-fuzzy equitable domination set of an anti-fuzzy graph is introduced. The connected end anti-fuzzy equitable domination number of an anti-fuzzy graphs is also obtained. 2. Preliminaries Definition 2.1[1] A fuzzy graph is said to be an anti-fuzzy graph with a pair of functions 𝜎 ∶ 𝑉 → [0,1] and 𝜇 ∶ 𝑉 × 𝑉 → [0,1], where for all 𝑢, 𝑣 ∈ 𝑉, we have 𝜇(𝑢, 𝑣) ≥ 𝜎(𝑢) ∨ 𝜎(𝑣) and it is denoted by 𝐺𝐴𝐹 (𝜎, 𝜇) or 𝐺(𝜎, 𝜇). Definition 2.2 [1] The order 𝑝 and size 𝑞 of an anti-fuzzy graph 𝐺𝐴𝐹 = (𝜎, 𝜇) are defined to be 𝑝 = ∑ 𝜎(𝑢)𝑢∈𝑉 and 𝑞 = ∑ 𝜇(𝑢, 𝑣)𝑢𝑣∈𝐸 . It is denoted by 𝑂(𝐺) and 𝑆(𝐺). Note 2.1 In all the examples, 𝜎 is chosen suitably and the function 𝜇 considered as reflexive and symmetric and 𝐺 is an undirected anti-fuzzy graph. Definition 2.3 An anti-fuzzy graph 𝐺𝐴𝐹 is said to be bipartite if the vertex set 𝑉 can be partitioned into two sets 𝜎1 on 𝑉1 and 𝜎2 on 𝑉2 such that 𝜇(𝑣1, 𝑣2) = 0 if (𝑣1, 𝑣2) ∈ 𝑉1 × 𝑉1 or (𝑣1, 𝑣2) ∈ 𝑉2 × 𝑉2. Definition 2.4 A bipartite anti-fuzzy graph 𝐺𝐴𝐹 is said to be complete bipartite anti- fuzzy graph if 𝜇(𝑣1, 𝑣2) = 𝜎(𝑣1) ∨ 𝜎(𝑣2) for all 𝑣1 ∈ 𝑉1 and 𝑣2 ∈ 𝑉2 and is denoted by 𝐾𝜎1,𝜎2 . Definition 2.5 A path in an anti-fuzzy graph 𝐺𝐴𝐹 is a sequence of distinct vertices 𝑢0, 𝑢1, 𝑢2, . . . , 𝑢𝑛 such that µ(𝑢𝑖−1 , 𝑢𝑖 ) = 𝜎(𝑢𝑖−1) ∨ 𝜎(𝑢𝑖 ), 1 ≤ 𝑖 ≤ 𝑛, 𝑛 > 0 is called the length of the path. The path in an anti-fuzzy graph is called an anti-fuzzy cycle if 𝑢0 = 𝑢𝑛, 𝑛 ≥ 3. 272 Connected end anti-fuzzy equitable dominating set in anti-fuzzy graphs Definition 2.6 An anti-fuzzy graph 𝐺𝐴𝐹 is said to be cyclic if it contains at least one anti-fuzzy cycle, otherwise it is called acyclic. Definition 2.7 An anti-fuzzy graph 𝐺𝐴𝐹 is said to be connected if there exists at least one path between every pair of vertices. A connected acyclic anti-fuzzy graph is said to be an anti-fuzzy tree. Definition 2.8 [4] Let 𝐺𝐴𝐹 be an anti-fuzzy graph and let 𝑢, 𝑣 ∈ 𝑉. If 𝜇(𝑢, 𝑣) = 𝜎(𝑢) ∨ 𝜎(𝑣) then 𝑢 dominates 𝑣 (or 𝑣 dominates 𝑢) in 𝐺𝐴𝐹 . A set 𝐷 ⊆ 𝑉 is said to be a dominating set of an anti-fuzzy graph 𝐺𝐴𝐹 if for every vertex 𝑣 ∈ 𝑉 − 𝐷 there exists 𝑢 ∈ 𝐷 such that 𝑢 dominates 𝑣. Definition 2.9 [4] A dominating set 𝐷 of an anti-fuzzy graph 𝐺𝐴𝐹 is called a minimal dominating set if there is no dominating set 𝐷′ such that 𝐷′ ⊂ 𝐷. Definition 2.10 [5] The maximum scalar cardinality taken over all minimal dominating set is called anti-fuzzy domination number of an anti-fuzzy graph 𝐺𝐴𝐹 and is denoted by 𝛾𝐴𝐹𝐺 𝑑 (𝐺𝐴𝐹 ). Definition 2.11 [2] Let 𝐺𝐴𝐹 be an anti-fuzzy graph. Let 𝑣1 and 𝑣2 be two vertices of 𝐺𝐴𝐹 . A subset 𝐷 of 𝑉 is called a anti-fuzzy equitable dominating set if every 𝑣2 ∈ 𝑉 − 𝐷 there exist a vertex 𝑣1 ∈ 𝐷 such that 𝑣1𝑣2 ∈ 𝐸 and |𝑑(𝑣1) − 𝑑(𝑣2)| ≤ 1 where 𝑑(𝑣1) denotes the degree of vertex 𝑣1 and 𝑑(𝑣2) denotes the degree of vertex 𝑣2 with 𝜇(𝑣1, 𝑣2) = 𝜎(𝑣1) ∨ 𝜎(𝑣2). Definition 2.12 [2] An anti-fuzzy equitable dominating set 𝐷 of an anti-fuzzy graph 𝐺𝐴𝐹 is called a minimal anti-fuzzy equitable dominating set if there is no anti-fuzzy equitable dominating set 𝐷′ such that 𝐷′ ⊂ 𝐷. The maximum scalar cardinality taken over all minimal anti-fuzzy equitable dominating set is called anti-fuzzy equitable domination number and is denoted by 𝛾𝐴𝐹𝐺 𝑒𝑑 . Definition 2.13 [4] An anti-fuzzy equitable dominating set 𝑆 of a connected anti-fuzzy graph 𝐺𝐴𝐹 is called the end anti-fuzzy equitable dominating set if 𝑆 contains all the terminal vertices. Definition 2.14 [4] The maximum scalar cardinality taken over all minimal end anti- fuzzy equitable dominating set is called end anti-fuzzy equitable domination number of 𝐺𝐴𝐹 and it is denoted by 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 . Definition 2.15 If for each 𝑥 ∈ 𝑉 − 𝑆 there exist a vertex 𝑦 ∈ 𝑆 such that 𝑥𝑦 ∈ 𝐸(𝐺𝐴𝐹) and either one of the vertex 𝑥 or 𝑦 is with degree 𝑘 and other vertex is with degree 𝑘 + 1, then 𝐺𝐴𝐹 is called a bi-regular anti-fuzzy graph. 3. Connected end anti-fuzzy equitable dominating set Definition 3.1 An end anti-fuzzy equitable dominating set 𝑆 of an anti-fuzzy graph 𝐺𝐴𝐹 is called the connected end anti-fuzzy equitable dominating set (CEAFED-set) if induced anti-fuzzy subgraph < 𝑆 > is connected. 273 S. Firthous Fatima and K. Janofer Definition 3.2 The maximum scalar cardinality taken over all minimal connected end anti-fuzzy equitable dominating set is called connected end anti-fuzzy equitable domination number of 𝐺𝐴𝐹 and it is denoted by 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑. Example 3.3 Consider the following anti-fuzzy graph 𝐺𝐴𝐹 , In the anti-fuzzy graph 𝐺𝐴𝐹 , given in figure 3.1, the minimal connected end anti-fuzzy equitable dominating sets are 𝑆1 = {𝑣3, 𝑣4, 𝑣5, 𝑣6, 𝑣7} and 𝑆2 = {𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6}. The scalar cardinality of 𝑆1 = |{𝑣3, 𝑣4, 𝑣5, 𝑣6, 𝑣7}| = 0.4 + 0.2 + 0.3 + 0.6 + 0.6 = 2.1 The scalar cardinality of 𝑆2 = |{𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6}| = 0.5 + 0.4 + 0.2 + 0.3 + 0.6 = 2 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 = max{|𝑆1|, |𝑆2|} = max{2.1, 2} = 2.1 𝑣6(0.6) 𝑣5(0.3) 0.6 0.4 𝑣7(0.6) 0.6 0.4 𝑣4(0.2) 𝑣3(0.4) 0.6 0.6 0.5 𝑣1(0.3) 0.5 𝑣2(0.5) Figure 3.1: Anti-fuzzy graph 𝐺𝐴𝐹 Therefore, connected end anti-fuzzy equitable domination number is 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 = 2.1 corresponding to the connected end anti-fuzzy equitable dominating set 𝑆1. Theorem 3.4 Let 𝐺𝐴𝐹 be any connected anti-fuzzy graph. Then 𝛾𝐴𝐹𝐺 𝑑 (𝐺𝐴𝐹 ) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). Proof: Let 𝐺𝐴𝐹 be any connected anti-fuzzy graph. Let 𝑆 ⊆ 𝑉(𝐺𝐴𝐹 ) be any 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 −set in 𝐺𝐴𝐹 . Then obviously, 𝑆 is also an end anti-fuzzy equitable dominating set in 𝐺. Therefore, 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) = |𝑆| ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) (1) Suppose let 𝑆′ be any 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 −set of 𝐺𝐴𝐹 . By definition of end anti-fuzzy equitable dominating set, 𝑆′ is also an anti-fuzzy equitable dominating set of 𝐺𝐴𝐹 . Therefore, 𝛾𝐴𝐹𝐺 𝑒𝑑 (𝐺𝐴𝐹 ) = |𝑆′| ≤ 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) (2) We know that, every anti-fuzzy equitable dominating set is an anti-fuzzy dominating set. Therefore, 𝛾𝐴𝐹𝐺 𝑑 (𝐺𝐴𝐹 ) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑑 (𝐺𝐴𝐹 ) (3) Hence from (1), (2) and (3), we have 274 Connected end anti-fuzzy equitable dominating set in anti-fuzzy graphs 𝛾𝐴𝐹𝐺 𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑑 (𝐺𝐴𝐹 ) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ). Remarks 3.5 The equality of theorem 3.4 can be hold when the anti-fuzzy graph 𝐺 has no isolated vertices. For example, anti-fuzzy cycle and complete anti-fuzzy graphs can hold the equality condition. Theorem 3.6 For any connected anti-fuzzy graph 𝐺𝐴𝐹 , 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). Proof: Let 𝑆 ⊆ 𝑉 be the minimum connected anti-fuzzy equitable dominating set of a connected anti-fuzzy graph 𝐺𝐴𝐹 . Then 𝑆 is an anti-fuzzy dominating set of 𝐺 and the induced anti-fuzzy subgraph < 𝑆 > is connected. Therefore, 𝑆 is also a connected anti-fuzzy dominating set. Clearly, any connected end anti-fuzzy equitable dominating set is also connected equitable dominating set. Hence, 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). Theorem 3.7 For any 𝑘-regular anti-fuzzy graph for 𝑘 > 1 then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ) Proof: Let us assume that 𝐺𝐴𝐹 be a k-regular anti-fuzzy graph. Then each vertex of 𝐺𝐴𝐹 has a same degree 𝑘. Let 𝑆 be the minimal connected anti-fuzzy equitable dominating set of 𝐺𝐴𝐹 Then cardinality of 𝑆 = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹). If 𝑢 ∈ 𝑉 − 𝑆 then 𝑆 is connected anti-fuzzy equitable dominating set, then there exists 𝑣 ∈ 𝑆 and 𝑢𝑣 be the effective edge, also 𝑑(𝑢) = 𝑑(𝑣) = 𝑘. Therefore |𝑑(𝑢) − 𝑑(𝑣)| = |𝑘 − 𝑘| = 0 ≤ 1. Hence 𝑆 is a connected end anti-fuzzy equitable dominating set of 𝐺 such that 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹 ) ≥ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) (1) By theorem 3.6, we have 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). (2) Hence, from (1) and (2), 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ) Corollary 3.8 Let 𝐺𝐴𝐹 be (𝑘, 𝑘 + 1) bi-regular anti-fuzzy graph. Then, 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺). Proof: By theorem 3.6, 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑(𝐺). Now, let 𝑆 be minimum connected end anti- fuzzy equitable set of (𝑘, 𝑘 + 1) bi-regular anti-fuzzy graph. By the definition, the connected end anti-fuzzy equitable dominating set 𝑆 is also an anti-fuzzy equitable 275 S. Firthous Fatima and K. Janofer dominating set and 〈𝑆〉 is connected, since 𝐺 is (𝑘, 𝑘 + 1) bi-regular anti-fuzzy graph. Therefore, 𝑆 is also a connected end anti-fuzzy equitable dominating set. Theorem 3.9 Let 𝐺𝐴𝐹 be an anti-fuzzy graph with 𝑛 vertices then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺) = 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺) if and only if 𝐺𝐴𝐹 has no end vertex and there is atleast one vertex 𝑣 ∈ 𝑉 adjacent to (𝑛 − 1) vertices in 𝐺𝐴𝐹 . Proof: Let 𝐺𝐴𝐹 be an connected anti-fuzzy graph with n vertices and without end vertex. Then, every vertex adjacent to atleast two vertices and there exists a one vertex say 𝑢 ∈ 𝑉(𝐺𝐴𝐹 ) adjacent to (𝑛 − 1) vertices, then the set 𝑆 = {𝑢} is connected end anti- fuzzy equitable dominating set 𝐺. By theorem 3.4, we get 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). Conversely, suppose 𝐺𝐴𝐹 is connected anti-fuzzy graph and 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) then 𝐺 has no end vertex and there is 𝑆 = {𝑣} which is connected end anti-fuzzy equitable dominating set. Therefore atleast any one vertex adjacent to (𝑛 − 1) vertices in 𝐺. Corollary 3.10 Let 𝐺𝐴𝐹 be an anti-fuzzy cycle with order 𝑝 then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) = 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}}. Proof: Since 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹 ) = 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}} and 𝑆 = 𝑉 – {𝑢, 𝑣} is any subset of the vertices on the anti-fuzzy cycle 𝐺𝐴𝐹 such that 𝑢 and 𝑣 are adjacent vertices. Clearly 𝑆 is connected end anti-fuzzy equitable set of 𝐺 it means 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}} and by the theorem 3.6, we have, 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}} = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ). Hence 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺) = 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}}. Theorem 3.11 For any complete bipartite anti-fuzzy graph then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹 ) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ) = { max 𝑢∈𝑉1 {𝜎(𝑢)} + max 𝑣∈𝑉2 {𝜎(𝑣)}, |𝑑(𝑢) + 𝑑(𝑣)| ≤ 1 𝑝 + 𝑞 , |𝑑(𝑢) + 𝑑(𝑣)| > 1 Proof: Case (1) : If 𝐺 ≅ 𝐾𝑚,𝑛 and |𝑑(𝑢) + 𝑑(𝑣)| > 1 for all 𝑢 ∈ 𝑉1 and 𝑣 ∈ 𝑉2 then the anti- fuzzy graph 𝐺𝐴𝐹 is totally anti-fuzzy equitable disconnected. Therefore, 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹 ) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) = 𝑝 + 𝑞. Case (2): If 𝐺 ≅ 𝐾𝑚,𝑛 and |𝑑(𝑢) + 𝑑(𝑣)| ≤ 1 then if 𝑉1 and 𝑉2 be the partite sets of 𝐺𝐴𝐹 , be selecting one vertex 𝑢 ∈ 𝑉1 and 𝑣 ∈ 𝑉2 then 𝑆 = {𝑢, 𝑣} is connected end anti- fuzzy equitable dominating set. 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝑚𝑎𝑥𝑢∈𝑉1 𝜎(𝑢) + 𝑚𝑎𝑥𝑣∈𝑉2 𝜎(𝑣) , but 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ) ≠ 𝑚𝑎𝑥𝑢∈𝑉 𝜎(𝑢). 276 Connected end anti-fuzzy equitable dominating set in anti-fuzzy graphs 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) = max 𝑢∈𝑉1 {𝜎(𝑢)} + max 𝑣∈𝑉2 {𝜎(𝑣)}. Theorem 3.12 Let 𝐺𝐴𝐹 be connected anti-fuzzy graph with order 𝑚 and 𝑁 be the set of all end anti- fuzzy vertices of 𝐺𝐴𝐹 then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) ≥ |𝑁| + max {𝜎(𝑥)}, where 𝑥 ∈ 𝑉 which is not an end anti-fuzzy vertex. Proof: Clearly if 𝐺𝐴𝐹 is connected anti-fuzzy graph with order 𝑚. Let 𝑁 be the set of all end anti-fuzzy vertices of 𝐺𝐴𝐹 . Let 𝑆 be any connected end anti- fuzzy equitable dominating set of 𝐺𝐴𝐹 then all the end vertices and also supporting vertices must along to end anti-fuzzy equitable dominating set. Hence 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) ≥ |𝑁| + max {𝜎(𝑥)}. 4 Conclusions A mathematical model helps to accomplish the problem in a complex situation. The possible solution is to convert the problem into a graph model. Anti-fuzzy graph theory has been used to model many decision making problems in uncertain situations. It have numerous applications in modern science in technology, computer science, especially in the field of information theory, neural network, cluster analysis, diagnosis and control theory etc., In this paper, the connected end anti-fuzzy equitable dominating set of an anti-fuzzy graph is defined. The relation between anti-fuzzy equitable domination number, end anti-fuzzy equitable domination number and connected end anti-fuzzy equitable domination number are established. Theorems related to these parameters are established. In future, we are going to establish these types of parameters in edge dominating sets of anti-fuzzy graphs. 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