2-anti fuzzy domination in anti fuzzy graphs

The aim of this paper is to study the concepts of the concepts of 2- anti fuzzy dominating (2 – AFD) set and 2- anti fuzzy domination number (2 – AF domination number) of an anti-fuzzy graph (GAF ).We determine the 2- anti fuzzy domination number (2 – AFD number γ 2AF ) for some classes of an anti-fuzzy graphs (G AF ) and obtain the bounds on (2 – AFD number) for the same. The relations between (2 – AF domination number), anti fuzzy domination (AF domination number) and vertex anti-fuzzy vertex covering number (α 0) are discussed and found some result.

: The aim of this paper is to study the concepts of the concepts of 2-anti fuzzy dominating (2 − ) set and 2-anti fuzzy domination number (2 − domination number) of an anti-fuzzy graph ( ).We determine the 2-anti fuzzy domination number (2 − number 2 ) for some classes of an anti-fuzzy graphs ( ) and obtain the bounds on (2 − number) for the same. The relations between (2 − domination number), anti fuzzy domination ( domination number) and vertex anti-fuzzy vertex covering number ( 0 ) are discussed and found some result. : Anti fuzzy graph, number, 2 − number and anti fuzzy vertex covering number 0 .
. The idea of domination came through the game of chess, to solve the problem of placing the fewer number of chess pieces to dominate of all square chess board. In 1962, Ore used crisp graphs to study domination theory [1]. The concept was developed by E. J. Cockayne and S. T. Hedetnieme [2]. The notion domination number plays an important role to solve various life problems by various kinds of fields in graph theory as topological graphs [3] and labeled graph [4], [5] and the others. Also, the researcher discussed many parameters of domination as in [6,7]. In 1965, L.
Zadeh [8] published his paper "fuzzy set "in which clarified the uncertainty. In (1975) Rosenfeld [9] introduced the concept of fuzzy graph .In (1975) Kauffmann [10] introduced the fundamental idea of a fuzzy graph. The concept of Domination in fuzzy graphs I, introduced by Somasundaram A. and Somasundaram S. [11,12] and them using effective edges. The concept of Domination in fuzzy graph, so introduced by Nagoorgani A. and Chandrasekaran V. T. [13] but by using strong edges.
Muhammad A. [14] was introduced the concept of Anti fuzzy structures graphs, and discussed the concepts connected anti fuzzy graphs, constant anti fuzzy graphs and other concepts. On anti fuzzy graph, Domination on anti fuzzy graph and connected domination on anti fuzzy are introduced by Muthuraj R. and Sasireka A. [15][16][17].In this paper we introduce the concept of 2-anti fuzzy

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We provide definition of anti fuzzy graph with a set of concepts related to it.
Let ⊆ × where a non-empty set and finite. Then = ( , ) is called anti fuzzy graph such that for every 1  respectively. [15]. A path P in is a sequence of distinct vertices 1 2 3 , , ,..., n x x x x such that ( 1 , 2 , … , −1 , ) > 0 and 1 ≤ ≤ and if there exist a fuzzy path between any two vertices then is connected [15]. well it is called maximal if there is no independent anti-fuzzy set * of such that | * | > | | [17].
The independence number 0 ( ) is the maximum fuzzy cardinality over all maximal independent anti fuzzy set of [19]. represented by 0 ( ) [11]. A vertex u is called end vertex of if it has exactly one effective neighbor in [12]. . − .
In this section, we introduce the 2 − , ≥ 4 , , ∈ 1 , ∈ 2 Proof. Given ≡ 1, 2 be a complete bipartite anti fuzzy graph of two bipartite sets 1 of vertices and 2 of vertices. So, there are three cases as follows.
Case 1. If or less than four, then there are two subcases as follows.  ,,   We see that  Proof: Let be 2 − such that | | = 2 of which, is not independent set, and suppose that ∈ such that = | | .We have two cases: Case1: If ( ) ∩ ( − ) = ∅ , since has no isolates vertex and is not independent then Proof: Let be a minimum anti fuzzy vertex cover set of G AF , and let ∈ (G AF ) − .
Clearly, ( ) ∈ . Since each vertex in G AF has at least two neighbors, the vertex is adjacent to at least two vertices of .This implies that is 2 − set of G AF . Hence, 2 ≤ 0 . □ then each vertex has at least two neighbors.

Proof:
Assume that there is a vertex has one neighbor, and let be a 2 − set of G AF such that | | = 2 , then ∈ , if − = ∅ then < 2 < Ῥ = 2 which is a contradiction,