Tikrit Journal of Pure

I n this paper, we define fuzzy graph chains, which comprise vertex identification. These fuzzy graphs are isomorphic fuzzy graphs, provide that after applying various features to the chain of fuzzy graphs, which as special fuzzy graph chain of 𝐶 5 .


Introduction
One application tool in the field of mathematics is the fuzzy graph, which enables users to simply explain the link between any two conceptions. The concept of the graph with fuzziness and many graph theory analogs in the fuzziness ideas such as paths, cycles, and connectedness were first described by Rosenfeld in 1975 [1]. In 2021, Mahmood and Ahmed introduced the vertex identification chain graphs " Schultz and Modified Schultz Polynomials for Vertex Identification Chain and Ring for Hexagon Graphs" [2]. Somasundaram introduced the concept of domination in graph with fuzziness in1998 [3].
Nagoorgani in 2007 introduced domination depending on the strong edges in fuzzy graphs [4]. In this paper we introduced the idea of a chain of fuzzy graphs and a new type of dominating set which is an equal dominating set. ∀ , ∈ , the relation ( , ) ≤ ( ) ∧ ( ) is satisfied [5]. path in graph with fuzziness is a collection of different vertices 0 , … , , where 0 ≠ , and ≥ 2 such that ( −1 , ) > 0, = 0, … , . We refer to the following pairs as the path's edges. Length of path is the number of edges [1]. A path where 0 = ≥ 3 is a cycle. The weight of the weakest edge ( the edge with least membership in path) is used to measure a path's strength. The strength of the connectedness between the vertices and , is the maximum strength of all paths linking them, and it is denoted by ( , ). A path connecting two vertices shows that they are linked [6].

The Vertex-Identification Chain Fuzzy Graphs:
The following is a formal definition of a chain of fuzzy graphs. Definition3.1.1: Assuming that { 1 , 2 , … , }, be a set of pairwise disjoint fuzzy graphs with vertices i , i ∈ ( i ) then the vertex-identification chain fuzzy graph , C vf ( 1 , 2 , … , n ≡ C vf ( 1 , 2 , … , n : 1 ⋅ n−1 is the graph obtained from the fuzzy graphs 1 , 2 , … , n by identifying the v i vertex with the vertex u i+1 for all i = 1,2 … n. such that the vertex identification's weight value is

Dominating set on Chain Fuzzy Graphs: Definition 3.2.1[3]: A vertex
dominates other vertices in a fuzzy graph = ( , , ). If ( , ) = ( ) ∧ ( ), for , ∈ . A set ⊆ is defined as a dominant set in if for each ∈ − , there exists ∈ , so that, dominates . The dominance number of is a collection with the least possible cardinality. and is denoted by (

Theorem 3.2.4: In a chain of fuzzy graphs if ≤ 6.
Then the set of the vertex identifications is a strong domination number. Proof: Let ( ) be a chain of fuzzy graphs with vertex identifications ∈ . Let ≤ 6. Let be the strongly dominant set. The degree of the vertex identification is greater than other vertices. Hence ≤ 6 then the vertex identification ∈ , is dominating other vertices strongly in − . The vertex identification is the minimal cardinality of a strong dominant set. Then the set of the vertex identifications is a strong domination number. Theorem 3.2.5: In a chain of fuzzy graphs if 4 ≤ ≤ 6. Then ( ( )) ≤ ( ( )).
Proof: Let ( ) be a chain of fuzzy graphs with vertex identification . Let 4 ≤ ≤ 6. The vertex identification is a strong dominance number, and it is the minimal number of vertices of a strong dominating set. In every fuzzy graph in the chain, there exist at least two weakly dominating vertex, which means the cardinality of a weak dominating set is greater than the cardinality of a strong dominating set.

Conclusions
In this paper, we discussed the relationship between a dominating set, a strong dominating set, and a weak dominating set on a chain of fuzzy graphs. We presented a new kind of dominating set called an equal dominating set. The vertex identification satisfies the conditions of dominating, strong dominating, and an equal dominating set.