Properties of Fuzzy Resolving Set

Asbract: In a fuzzy graph G(v, σ, μ), for a subset H of σ, the representation of σ − H with respect to H in terms of strength of connectedness of vertices are distinct then H is called the fuzzy resolving set of G. In this article, we discuss the properties of fuzzy resolving set and fuzzy resolving number. And also proved some theorems and properties of fuzzy resolving number in fuzzy cyclic and fuzzy labeling graphs.


Introduction
Lotfi Asker Zadeh explained the Fuzzy Mathematics in 1965, later in the year 1975, Fuzzy Graph theory is introduced by Rosenfield. The fuzzy graph is a current research area and it has plenty of applications in real life. Therefore a lot of researchers do research in fuzzy graph. There are different types of fuzzy graphs namely fuzzy labeliing graphs, fuzzy cyclic graphs, intustiastic fuzzy graph, complete fuzzy graphs, irregular fuzzy graphs, regular fuzzy graphs etc.. The concept of colouring, labeling, domination, fuzzy clique etc., are the research areas which are more demanding. Slater in 1975, explained the concept of Resolving number of a graph. It is applied to find the position of the robot differently in a graph-structured framework. If we assume that the robots are moving in a fuzzy graph-structured framework and it needs to identify its position uniquely in terms of safety level, this motivated Shanmugapriya and Jiny to introduced the fuzzy resolving set, fuzzy resolving number, fuzzy super resolving set and a fuzzy super resolving number in the year 2019. As an extension of the research work, we will discuss some properties of resolving number of a fuzzy graph in this paper.

Definitions
Definition 2.1 Let be a non-empty set of vertices, is the function from set of all vertices to [0,1] and is a function from × to [0,1] such that for all , ∈ , ( , ) ≤ ( ) ∧ ( ), then the ordered triple ( , , ) is called the fuzzy graph. The support of and are given as * = {( , )/ ( , ) > 0} and * = { / ( ) > 0}. [2]. Definition 2.2 Strength (or weight) of the path = 1 , 2 , . . . is the membership value of the weakest edge in the path . The path is called a cycle if 1 = .

Definition 2.3
The weight (or strength)of connectedness between the two vertices 1 to is the maximum of strength of all paths between 1 to , is denoted as ∞ ( 1 , ) [3], for simplicity of our usage, we use ( 1 , ).

Definition 2.5 A 2-dominating set
is a subset of such that for all ∈ − there exist minimum two strong neighbour in . The minimum cardinality of a 2-dominating set is called 2-domination number of the fuzzy graph and is denoted as 2 ( ). If the representation of with respect to are distinct then is called the fuzzy super resolving set of . The cardinality of the minimum super resolving set is called the fuzzy super resolving number of , denoted as Sr(G). [1].

Definition 2.7 The Connectedness matrix
of the fuzzy graph is defined as = ∞ ( , ) for ≠ and = 0 if = .
Result 1: If is a fuzzy resolving set of a connceted fuzzy labeling graph , then is a fuzzy super resolving set of . [4].
Result 2: Every fuzzy labeling cycle of ′2 − 1′ vertices such that * is a cycle, has a fuzzy resolving set of cardinality ′ − 1′.
Note 1: through out this paper we consider a fuzzy graph with | | ≤ 3. Note 2: For any fuzzy graph , 2 ≤ | | ≤ − 1. The representation of − 1 with respect to 1 are all distinct, therefore 1 is the resolving set of . Similarly we can see that, 2 = { 1 , 3 }, 3 = { 1 , 4 }, 4 = { 2 , 3 }, 5 = { 2 , 4 } are all fuzzy resolving set of and 6 = { 3 , 4 } is not a fuzzy resolving set of . Hence, the minimum resolving set has cardinality 2, that is, ( ) = 2. is a strong arc for all ∈ and is a fuzzy labeling.]. Therefore is a fuzzy resolving set and which is also a fuzzy super resolving set by result [1]. Hence ( ) = ( ) = 2.  Then the corresponding edges will be adjacent in at-least one place. Let it be and . choose the non adjacent vertices and and then choose any one of the two remaining vertices from − { , , }, these three vertices will form the crisp set * ∈ of the resolving set ∈ (the representation of − with respect to will be different). That is there always exist a resolving set of with cardinality three. If we consider any two element subset of whcih will not form a resolving set since there exist only two distinct edge membership value, out of the three representation of − with respect to any two will be the same. And hence from the definition of fuzzy resolving number, the fuzzy resolving number of in this case is 3.

Case(ii) If there is exactly three different edge weight in .
Then the three edges with different edge membership value will be consecutive in at least one place of the cycle * which will form a path. If we have only one such path then choose the two end vertices of the path for * ∈ and the corresponding ∈ will form resolving set of cardinality 2. And therefore by definition, the fuzzy resolving number of is 2.
Case(ii)(a) if we have two such paths and there exist only one weakest edge then choose any one of path, then select the one end vertex of the edge with maximum membership value in the path and choose one end vertex of the edge with minimum membership value which is not adjacent to the chosen vertex for * ∈ and the corresponding ∈ will form resolving set of cardinality 2.
Case(ii)(b) if we have two such paths and there is two weakest edge (if a cyclic graph has more than one weakest arc then it is called fuzzy cycle) which are not adjacent then choose the end vertex of the any one path and choose any one of the remaining vertex for * ∈ and the corresponding ∈ will form resolving set of cardinality 3.
Case(ii)(c) if we have two such paths and there is two weakest edge which are adjacent. Let 1 < 2 < 3 are the three different edge membership value. Now select one vertex of the edge with membership value 2 and the vertex of the edge with membership value 3 and not adjacent to the selected one for * ∈ and the corresponding ∈ will form resolving set of cardinality 2.

Case(iii) If there is exactly four different edge weight
Then the corresponding edges will be consecutive in the cycle and which will form a path. then follow the case (ii) to get a resolving set of cardinality 2.
Case (iv) if all five edges have a different edge membership value.
Then it will form a fuzzy labelling cycle of order 5. then by result [2] the resolving number of is 2.
By case (i), (ii),(iii) and (iv) the resolving number of is either 3 or 2.

Fig. 2. Cyclic Fuzzy Graphs
Theorem 3.5 Let ( , , ) be a fuzzy graph with | | = . If is a resolving set of with | | = /2 then − is does not need to be a resolving set of .
proof: By definition [2.7], the conncectedness matrix of the fuzzy graph is a symmetric matrix. let is a subset of with | | = /2. If we put the representation of − with respect to in the row of a 2 × 2 matrix say 1 and the representation of with respect to − in the row of a 2 × 2 matrix say 2 . Then 1 and 2 will be transpose to each other[ 1 = 2 ]. Now let is the resolving set of then each row of 1 are distinct which implies that each column of 2 are distinct. And which does not implies that each row of 2 are distinct. Therefore the representation of with respect to − is does not need to be distinct. Hence − does not need to be a resolving set of .

Conclusion
Resolving set of a fuzzy graph is a trending topic in fuzzy graph. In this paper, we have discussed some properties of resolving set of the fuzzy graph. We have proved that the resolving number of fuzzy graph with * is a four cycle is 2 and five cycle is 2 or 3. We have also proved that for a fuzzy labeling graph if the 2dominating number is 2 then the resolving number of is also 2. We would like to work on finding the relation between the domination set of the fuzzy graph and the resolving set in future.