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Applied Computational Intelligence and Soft Computing
Volume 2016, Article ID 5859080, 13 pages
http://dx.doi.org/10.1155/2016/5859080
Research Article

Application of Bipolar Fuzzy Sets in Graph Structures

Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan

Received 27 November 2015; Revised 25 December 2015; Accepted 28 December 2015

Academic Editor: Baoding Liu

Copyright © 2016 Muhammad Akram and Rabia Akmal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A graph structure is a useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. In this paper, we apply the concept of bipolar fuzzy sets to graph structures. We introduce certain notions, including bipolar fuzzy graph structure (BFGS), strong bipolar fuzzy graph structure, bipolar fuzzy -cycle, bipolar fuzzy -tree, bipolar fuzzy -cut vertex, and bipolar fuzzy -bridge, and illustrate these notions by several examples. We study -complement, self-complement, strong self-complement, and totally strong self-complement in bipolar fuzzy graph structures, and we investigate some of their interesting properties.

1. Introduction

Concepts of graph theory have applications in many areas of computer science including data mining, image segmentation, clustering, image capturing, and networking. A graph structure, introduced by Sampathkumar [1], is a generalization of undirected graph which is quite useful in studying some structures including graphs, signed graphs, and graphs in which every edge is labeled or colored. A graph structure helps to study the various relations and the corresponding edges simultaneously.

A fuzzy set, introduced by Zadeh [2], gives the degree of membership of an object in a given set. Zhang [3] initiated the concept of a bipolar fuzzy set as a generalization of a fuzzy set. A bipolar fuzzy set is an extension of fuzzy set whose membership degree range is . In a bipolar fuzzy set, the membership degree of an element means that the element is irrelevant to the corresponding property, the membership degree of an element indicates that the element somewhat satisfies the property, and the membership degree of an element indicates that the element somewhat satisfies the implicit counterproperty. Kauffman defined in [4] a fuzzy graph. Rosenfeld [5] described the structure of fuzzy graphs obtaining analogs of several graph theoretical concepts. Bhattacharya [6] gave some remarks on fuzzy graphs. Several concepts on fuzzy graphs were introduced by Mordeson et al. [7]. Dinesh [8] introduced the notion of a fuzzy graph structure and discussed some related properties. Akram et al. [913] have introduced bipolar fuzzy graphs, regular bipolar fuzzy graphs, irregular bipolar fuzzy graphs, antipodal bipolar fuzzy graphs, and bipolar fuzzy hypergraphs. In this paper, we introduce the certain notions including bipolar fuzzy graph structure (BFGS), strong bipolar fuzzy graph structure, bipolar fuzzy -cycle, bipolar fuzzy -tree, bipolar fuzzy -cut vertex, and bipolar fuzzy -bridge and illustrate these notions by several examples. We present -complement, self-complement, strong self-complement, and totally strong self-complement in bipolar fuzzy graph structures, and we investigate some of their interesting properties.

We have used standard definitions and terminologies in this paper. For other notations, terminologies, and applications not mentioned in the paper, the readers are referred to [1, 5, 7, 1418].

2. Preliminaries

In this section, we review some definitions that are necessary for this paper.

A graph structure consists of a nonempty set together with relations on , which are mutually disjoint such that each is irreflexive and symmetric. If for some , we call it an -edge and write it as “.” A graph structure is complete, if (i) each edge , appears at least once in ; (ii) between each pair of vertices in , is an -edge for some A graph structure is connected, if the underlying graph is connected. In a graph structure, -path between two vertices and , is the path which consists of only -edges for some , and similarly, -cycle is the cycle which consists of only -edges for some . A graph structure is a tree if it is connected and contains no cycle or equivalently the underlying graph is a tree. is an -tree, if the subgraph structure induced by -edges is a tree. Similarly, is an -tree, if is an -tree for each A graph structure is an -forest, if the subgraph structure induced by -edges is a forest, that is, if it has no -cycles. Let ; then the subgraph structure induced by has vertex set , where two vertices and in are joined by an -edge, , if and only if, they are joined by an -edge in . For some , the -subgraph induced by is denoted by - It has only those -edges of , joining vertices in . If is a subset of edge set in , then subgraph structure induced by has the vertex set, “the end vertices in ”, whose edges are those in Let and be graph structures. Then and are isomorphic, if (i) , (ii) there exist a bijection and a bijection , say , , such that for all , implies that .

Two graph structures and , on the same vertex set , are identical, if there exists a bijection , such that for all and in is an -edge in , then is an -edge in , where and . Let be a permutation on . Then the -cyclic complement of , denoted by , is obtained by replacing by ), . Let be a graph structure and a permutation on ; then(i) is -self complementary, if is isomorphic to ; the -cyclic complement of and is self-complement, if identity permutation.(ii) is strong -self complementary, if is identical to ; the -complement of and is strong self-complement, if identity permutation.

Definition 1 (see [2]). A fuzzy subset on a set is a map . A fuzzy binary relation on is a fuzzy subset on . By a fuzzy relation we mean a fuzzy binary relation given by .

Definition 2 (see [8]). Let be a graph structure and let be the fuzzy subsets of , respectively, such thatThen is a fuzzy graph structure of .

Definition 3 (see [8]). Let be a fuzzy graph structure of a graph structure Then is a partial fuzzy spanning subgraph structure of if for

Definition 4 (see [8]). Let be a graph structure and let be a fuzzy graph structure of . If , then “” is said to be a -edge of .

Definition 5 (see [8]). The strength of a -path of a fuzzy graph structure is for .

Definition 6 (see [8]). In a fuzzy graph structure , , , for any Also .

Definition 7 (see [8]). Let be a -edge of Let be a partial fuzzy spanning subgraph structure obtained by deleting “” with and -edges other than If for some supp, then is a -bridge.

Definition 8 (see [8]). Let be the partial fuzzy subgraph structure obtained by deleting vertex of , that is, and and Then a vertex of is a -cut vertex if for some with

Definition 9 (see [8]). is a -cycle if and only if is a -cycle.

Definition 10 (see [8]). is a fuzzy -cycle if and only if is an -cycle and there exists no unique “” in supp such that .

Definition 11 (see [8]). is a fuzzy -tree if it has a partial fuzzy spanning subgraph structure, , which is a -tree where for all -edges not in

Definition 12 (see [8]). Let be a graph structure and let be the fuzzy subsets of , respectively, such thatThen is a fuzzy graph structure of

Definition 13 (see [3]). Let be a nonempty set. A bipolar fuzzy set in is an object having the form where and are mappings.

We use the positive membership degree to denote the satisfaction degree of an element to the property corresponding to a bipolar fuzzy set and the negative membership degree to denote the satisfaction degree of an element to some implicit counterproperty corresponding to a bipolar fuzzy set . If and , it is the situation that is regarded as having only positive satisfaction for . If and , it is the situation that does not satisfy the property of but somewhat satisfies the counter property of . It is possible for an element to be such that and when the membership function of the property overlaps that of its counterproperty over some portion of .

For the sake of simplicity, we will use the symbol for the bipolar fuzzy set:

Definition 14 (see [3]). Let be a nonempty set. Then we call a mapping a bipolar fuzzy relation on such that and .

Definition 15 (see [9]). A bipolar fuzzy graph is a nonempty set together with a pair of functions and such that for all , Notice that , for , for , and is symmetric relation.

3. Bipolar Fuzzy Graph Structures

Definition 16. is called a bipolar fuzzy graph structure (BFGS) of a graph structure (GS) if is a bipolar fuzzy set on and for each ; is a bipolar fuzzy set on such that Note that for all and , , where and are called underlying vertex set and underlying -edge set of , respectively.

Definition 17. Let be a bipolar fuzzy graph structure of a graph structure If is a bipolar fuzzy graph structure of such that then is called a bipolar fuzzy subgraph structure of BFGS .
BFGS is a bipolar fuzzy induced subgraph structure of , by a subset of if Similarly, BFGS is a bipolar fuzzy spanning subgraph structure of if and

Example 18. Consider a graph structure such that , and
(i) Let , and be bipolar fuzzy subsets of , and , respectively, such that Then, by direct calculations, it is easy to see that is a BFGS of as shown in Figure 1.
(ii) Consider , , and . Then, by routine calculations, it is easy to see that is the bipolar fuzzy subgraph structure of as shown in Figure 2.

Figure 1: .
Figure 2: Bipolar fuzzy subgraph structure .

Definition 19. Let be a bipolar fuzzy graph structure of a graph structure Then is called a bipolar fuzzy -edge or simply -edge, ifThen support of , , consequently, is

Definition 20. -path in a BFGS of a graph structure is a sequence of distinct vertices (except the choice )in , such that is a bipolar fuzzy -edge for all

Definition 21. A BFGS with underlying vertex set is said to be -strong for some if for all A BFGS is said to be strong if it is -strong BFGS for all

Example 22. Consider BFGS as shown in Figure 3.
Then is a strong BFGS since it is both - and -strong.

Figure 3: BFGS .

Definition 23. A BFGS with underlying vertex set is said to be complete or -complete, if the following are true: (i) a is strong BFGS.(ii).(iii)For each pair of vertices is an -edge for some .

Example 24. Let be BFGS of graph structure such that , and as shown in Figure 4. By routine calculations, it is easy to see that is a strong BFGS.
Moreover, , and every pair of vertices belonging to is either an -edge or an -edge. So is a complete BFGS, that is, -complete BFGS.

Figure 4: .

Definition 25. Let be a BFGS with underlying vertex set Then positive and negative strengths of a -path” are called gain and loss of that -path and denoted by and , respectively, such that

Example 26. Consider a BFGS as shown in Figure 4. We note that is an -path. So . Consider

Definition 27. Let be a BFGS with underlying vertex set Then(i)-gain of connectedness between and is defined by , such that for and , where (ii)-loss of connectedness between and is defined by , such that for and , where

Example 28. Let be BFGS of graph structure such that , , and , as is shown in Figure 5.
Since , , , , , , and , thereforeSimilarly,This implies thatSincewe haveSimilarly,This implies thatFor all the remaining pairs of vertices, -loss and -gain of connectedness are zero.