Fuzzy Detour Convexity and Fuzzy Detour Covering in Fuzzy Graphs

A path P connecting a pair of vertices in a connected fuzzy graph is called a fuzzy detour, if its μ length is maximum among all the feasible paths between them. In this paper we establish the notion of fuzzy detour convex sets, fuzzy detour covering, fuzzy detour basis, fuzzy detour number, fuzzy detour blocks and investigate some of their properties. It has been proved that, for a complete fuzzy graph G, the set of any pair of vertices in G is a fuzzy detour covering. A necessary and sufficient condition for a complete fuzzy graph to become a fuzzy detour block is also established. It has been proved that for a fuzzy tree there exists a nested chain of sets, where each set is a fuzzy detour convex. Application of fuzzy detour covering and fuzzy detour basis is also presented.


Introduction
It was Lotfi A. Zadeh in 1965 [12] presented the notion of fuzzy sets, which deals with vagueness and lack of exactness in set theory. Later, Prof. Azriel Rosenfeld instilled fuzzy set theory into fuzzy graph theory in 1975 [11]. Infusing real world experience into fuzzy graph will make this area more rejuvenating. Rosenfeld's papers encouraged researchers to think differently and gave them freedom to explore. In fuzzy graphs, he developed fuzzy relations on fuzzy sets and verified graph theoretic concepts. Several other pioneering works in this field can be found in [6] [7] [8].
The notion of detour distance in graphs was presented by Gary Chartrand, Gamy L. Johns and Songlin Tian in [10]. It was Rosenfeld who presented a metric called thedistance [11] in fuzzy graphs, later, which was effectively used by various authors in their studies. Dr.A.Nagoorgani and J. Umamaheswari presented the notion of fuzzy detourdistance and some of its properties in [1]. Linda and Sunitha extensively studied the notions of -convex sets and its related properties in [3]. This paper serves three purposes. The first one is to establish the notion of fuzzy detour convex set, detour covering, detour basis, detour number and illustrate them with suitable examples. Second purpose is to present the concept of fuzzy detour blocks and some of their characteristics. Finally, to present an application of fuzzy detour covering and fuzzy detour basis.

Definition 2.6
Suppose that : ( , , ) be a connected fuzzy graph. The fuzzy detourdistance ∆( , ) between pair of vertices u and v is defined by the maximum -length of any uv path, where the -length of a path P: = 0, 1 ……. = is A uv path is called a uv fuzzy detour if its length is ∆( , ). Proposition 2.7 For the fuzzy detourdistance ∆ on a connected fuzzy graph : ( , , ), then ( ( ), ∆ ) is a metric space. Proposition 2.8 Every fuzzy detour in a complete fuzzy graph is a Hamiltonian path. Proposition 2.9 A complete fuzzy graph with n vertices has at least ( −1) 2 distinct fuzzy detour paths.

Fuzzy Detour Convexity
In fuzzy graph the concept of geodesic convexity and its related properties depending ondistance are introduced in [3]. In this section we establish the notion of fuzzy detour convexity based on fuzzy detourdistance [1] and present the notion of fuzzy detour covering, detour basis, detour number and some of their properties.
Definition 3.1 Let and be any two vertices in a connected fuzzy graph : ( , , ), the fuzzy detour closed set is defined to be the set that contains all the vertices in allfuzzy detour together with and and is denoted by [ , ]. Here the path 1 -3 -2 is a 1 -2 fuzzy detour. The edge 1 3 is a 1 -3 fuzzy detour. Thus Proposition 3.5 The null set ∅, every singleton sets in σ * and the whole vertex set σ * for a connected fuzzy graph : ( , , ) are fuzzy detour convex.
Note 1: If the order of a fuzzy detour convex set S varies over the range 2 ≤ ( ) < ( * ), then such set is said to be a nontrivial fuzzy detour convex set. Theorem 3.6 Suppose : ( , , ) be a connected fuzzy graph, then the intersection of any two fuzzy detour convex sets is also fuzzy detour convex.
Proof: Assume that : ( , , ) to be a connected fuzzy graph. Consider any two fuzzy detour convex sets of σ * , let it be U and W. We show that U ∩ W is fuzzy detour convex. Let and be any two vertices in U ∩ W. Therefore both and are in U and W. Also since U and W are fuzzy detour convex, we have all vertices in all fuzzy detours are both in U and W. Therefore we have all vertices in allfuzzy detours are in ∩ . Hence we are done.
Theorem 3.8 Suppose : ( , , ) be a fuzzy tree and let ′ : ( , * , * ) is a tree. Then we can find a nested chain of sets where is fuzzy detour convex for each i.
Proof: Suppose : ( , , ) be a fuzzy tree and graph ′ : ( , * , * ) is a tree with n vertices. Since ′ is a tree, assures the existence of only one path from one vertex to another vertex. Hence there exist only one fuzzy detour path between every pair of vertices, thus the union of all vertices in all detour path is equal to * = , which is fuzzy detour convex and order of equals n. Let 1 be a subgraph of induced by set −1 , obtained by deletion of a pendant vertex from . Clearly the induced graph 1 is a tree. By similar arguments mentioned above, −1 is a fuzzy detour convex and order of −1 equals n -1. Proceeding like this till we get a singleton set.
From the above example 3.14 we have the following observation, that for a complete fuzzy graph : ( , , ), the set { , } ∀ , * is a fuzzy detour covering of . This property is generalized from the next theorem. Proof: Let : ( , , ) be a complete fuzzy graph. Let , be any pair of vertices in * . We show that { , } is a fuzzy detour covering. Since is complete, by Proposition 2.8, every fuzzy detour path of a complete fuzzy graph is Hamiltonian [2]. This means, every fuzzy detour traverses through every vertex of exactly once. Therefore, for any pair of vertices and , we have [{ , }] = * . This proves that { , } is fuzzy detour covering for any pair of vertices , * . we are done. So we assume that there exist a fuzzy detour covering of , say , which is minimal than { , } for any pair of , * . Then should be a singleton set. Thus [ ] ≠ * , which is a contradiction. Therefore { , } is a minimal fuzzy detour covering of for any pair of vertices , * . Thus a basis of .
The next corollary is an instant consequence of above corollary 3.16 Corollary 3.17 Fuzzy detour number, ( ) for a complete fuzzy graph : ( , , ) of n vertices equals 2.

Applications
An application which is closely related is in the field of power supply across wide area. Demand of power can categorise over a particular time period, for example daily, seasonal, occasionally and so on. If we consider a particular city, the energy demand may be different in different areas depending on consumption. Demands may fluctuate on daily, monthly or yearly basis. Usually we prefer to distribute electric energy depending on the demand across different parts effectively and without huge loss of power.
A power station requires an additional electric power supply to meet the demand without failure for a certain time period. But station authorities doesn't have enough time to contact all other power stations. But at the same time authorities were forced for a rapid action in collaboration with all the neighbouring power stations for an additional power supply. Here we can solve this problem by using the concept of fuzzy detour covering. For that we consider a connected fuzzy graph. In particular consider all power stations within a city. Representing the power stations by vertices and transmission lines between them by edges. Every edge there is associated with a membership value (say, resistance level). The reciprocal of membership value represents the flow of current. If an edge have minimum membership value, then the flow of current through it will become high. Power stations may or may not be connected directly all together. More precisely if one power station does not connected the other, then there is no edge correspondence between them.
Let us assume that suddenly the power station 6 lacking enough power to meet the demand, so it requires emergency power supply from its neighbouring power stations. Now we consider all detour paths between 6 and other power stations.
We have path 1 : 6 -4 -5 − 3 − 2 − 1 is a fuzzy 6 -1 detour path and ∆( 6 , 1 ) = 20.83 2 : 6 -4 -5 − 3 − 1 − 2 is a fuzzy 6 -2 detour path and ∆( 6 , 2 ) = 22 3 : 6 -4 -5 − 2 − 1 − 3 is a fuzzy 6 -3 detour path and ∆( 6 , 3 ) = 15.3 4 : 6 -5 -2 − 1 − 3 − 4 is a fuzzy 6 -4 detour path and ∆( 6 , 4 ) = 14.8 5 : 6 -2 -1 − 3 − 4 − 5 is a fuzzy 6 -5 detour path and ∆( 6 , 5 ) = 14.8 Here { 6 , 1 }, { 6 , 2 }, { 6 , 3 }, { 6 , 4 } and { 6 , 5 } are fuzzy detour covering also they are basis for above fuzzy graph. In fuzzy detour path 2 has maximum flow rate, 4 and 5 has minimum flow rate as compared to other detour paths. So from above details we can conclude that power station 6 only to contact any one of other power station for additional power supply. So that power station those who receive message from 6 will charge the appropriate detour transmission line depending on demand and pass message to subsequent power stations in this line. In particular, power station 6 wants huge load to meet demand, then will opt path 2 . So power station 6 will send request to 2 for assistance. Thus { 2 , 6 } covers all power stations and ensure the required power supply for 6 . This process can be extended to a wide network. Furthermore, consider every detour distance between all pair of vertices within a particular network, from that find out all detour basis. This detour basis will play an important role in connecting and sharing facilities from one network to another network.
A complete fuzzy graph, fuzzy detour covering and fuzzy detour basis in association with applications of blockchain [13] based solution can considered to be appropriate for reducing intermediaries. Blockchains can broadly be defined as a new form of network infrastructure in the field of information technology that create 'trust' in networks by introducing distributed verifiability, higher level auditability, and general agreements.
Block chain act like a shared database and thus ensure trust across peer to peer network and also that no individual body can claim the ownership of blockchain network. Moreover no single individual can modify data stored on it or take decision solitarily without the permission from its peers. Here we can apply the application of fuzzy detour covering in blockchain. This may be explored in upcoming work.

Conclusion
In this paper, we presented the notions of fuzzy detour convex set, detour covering, detour basis and detour number for a fuzzy graph and illustrated them with suitable examples. It has been proved that the intersection of two fuzzy detour convex sets is also a fuzzy detour convex and set of each pair of vertices in a complete fuzzy graph is found to be a detour covering. This paper identifies that the fuzzy detour number for a complete fuzzy graph equals two and for a fuzzy tree there exists a nested chain of sets, where each set is a fuzzy detour convex. Finally, we presented an application of fuzzy detour covering and fuzzy detour basis in electric power transmission.