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

Research Article

Application of Bipolar Fuzzy Sets in Graph Structures

Muhammad Akram and Rabia Akmal

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. [9–13] 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, 14–18].

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 that

Then 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 that

Then 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, if

Then 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 , therefore

Similarly,

This implies that

Since

we have

Similarly,

This implies that

For all the remaining pairs of vertices, -loss and -gain of connectedness are zero.

Figure 5: .

Definition 29. A BFGS of a graph structure is an -cycle if is an -cycle.

Definition 30. A BFGS of a graph structure is a bipolar fuzzy -cycle for some if

(i) is an -cycle;(ii)there is no unique -edge in such that or .

Example 31. Consider BFGS as shown in Figure 3. Then is an -cycle as well as bipolar fuzzy -cycle, since is an -cycle and there are two -edges with minimum positive degree and more than one -edge with maximum negative degree of all -edges.

Definition 32. Let be a BFGS of a graph structure and a vertex of . Let be a bipolar fuzy subgraph structure of induced by such that

Then is a bipolar fuzzy -cut vertex for some , if

And, is an -P bipolar fuzzy cut vertex if only the first condition holds and a -N bipolar fuzzy cut vertex if only the second condition holds.

Example 33. Consider BFSG as considered in Example 28 and shown in Figure 5; after deleting vertex , the resulting bipolar fuzzy subgraph structure will be as shown in Figure 6.
Then is a bipolar fuzzy - cut vertex since

Figure 6: Bipolar fuzzy subgraph structure .

Definition 34. Let be a BFGS of a graph structure and let be an -edge. Let be a bipolar fuzzy spanning subgraph structure of , obtained by taking

Then is a bipolar fuzzy -bridge if

Edge is an -P bipolar fuzzy bridge if only the first condition holds and an -N bipolar fuzzy bridge if only the second condition holds.

Example 35. Consider the BFGS as shown in Figure 6 and let be bipolar fuzzy spanning subgraph structure of obtained by deleting -edge . Then is a bipolar fuzzy -bridge, since and , and also abipolar fuzzy -N bridge, since and

Definition 36. A BFGS of a graph structure is an -tree if is an -tree. In other words, is an -tree if a subgraph of , induced by , forms a tree.

Definition 37. A BFGS of a graph structure is a bipolar fuzzy -tree if has a bipolar fuzzy spanning subgraph structure such that is a -tree and and -edges not in
In more concerned view, is a bipolar fuzzy -P tree if only the first condition holds and a bipolar fuzzy -N tree if only the second condition holds.

Example 38. Consider BFGS as shown in Figure 7, which is an -tree. It is not an -tree but a bipolar fuzzy -tree since it has a bipolar fuzzy spanning subgraph structure as an -tree, which is obtained by deleting -edge from and

Figure 7: .

Definition 39. A BFGS of graph structure is isomorphic to a BFGS of if there exists a bijection and a permutation on the set such that

and for

Example 40. Let and be two BFGSs of graph structures and , respectively, as shown in Figure 8.
Here is isomorphic (not identical) to under the mapping , defined by , , and , and a permutation given by , , such that

Figure 8: Isomorphic bipolar fuzzy graph structures.

Definition 41. A BFGS of GS is identical to a BFGS of GS if there exist a bijection , such that

Example 42. Let and be two BFGSs of graph structures and , respectively, as shown in Figure 9.
Here is identical with under the mapping , defined by , and , such that

Figure 9: Identical bipolar fuzzy graph structures.

Definition 43. Let be a BFGS of a GS Let be any permutation on the set and the corresponding permutation on ; that is, if and only if
If for some and

then , while is chosen such that and
And BFGS , denoted by , is called the -complement of BFGS

Example 44. Let , , , , and be bipolar fuzzy subsets of , and , respectively, so that is a BFGS of graph structure Let be a permutation on the set such that and
Now for ,

Clearly, and So .
Similarly for , and
and . So .
And for , and
and . So .
This implies that

and is the -complement of

Theorem 45. A -complement of a bipolar fuzzy graph structure is always a strong BFGS. Moreover, if for , then all -edges in BFGS become -edges in

Proof. From the definition of -complement ,

for
Let us consider expression (37) first.
Since and , we can write

Also from the definition of a BFGS

Therefore, .
Now a requirement is minimum value of Since , that is why it is minimum when its positive part is zero. And when and is an -edge. So

Similarly for expression (38), a requirement is maximum value of Since , and

Therefore, .
Now will be maximum when its negative part becomes zero. Clearly, when and is an -edge. So

From (41) and (43), the conclusion is obvious.

Definition 46. Let be a BFGS and let be any permutation on the set Then

(i) is self-complement if it is isomorphic to , the -complement of ;(ii) is strong self-complement if it is identical to .

Definition 47. Let be a BFGS. Then

(i) is totally self-complement if it is isomorphic to , the -complement of , for all permutations on the set ;(ii) is totally strong self-complement if it is identical to , the -complement of , for all permutations on the set

Example 48. All strong BFGSs are the only examples of self-complement or totally self-complement BFGSs.

Example 49. A BFGS of graph structure as shown in Figure 10 is totally strong self-complement.

Figure 10: Totally strong self-complement BFGS.

Theorem 50. A BFGS is strong if and only if it is totally self-complement.

Proof. Let be a strong BFGS and any permutation on the set .
By Theorem 45, is strong and if , then all -edges in become -edges in

Hence is isomorphic to under the identity mapping , such that and

for This holds for any permutation on the set
Hence is totally self-complement.
Conversely, let and be isomorphic for any permutation on the set Then from the definition of -complement and isomorphism of BFGSs, we have

.
Hence, is a strong BFGS.

Remark 51. Every self-complement BFGS is necessarily totally self-complement.

Theorem 52. If graph structure is totally strong self-complement and is a bipolar fuzzy set of with constant valued functions and , then a strong BFGS of is totally strong self-complement.

Proof. Let and be two constants, such that

Since is totally strong self-complement, so for every permutation on the set , there exists a bijection , such that for every -edge , “” [an -edge in ] is an -edge in and, consequently, for every -edge , “” [a -edge in ] is a -edge in Moreover is strong, so we have

.
This shows that is strong self-complement. This holds for any permutation and on the set ; thus is totally strong self-complement. This completes the proof.

Remark 53. The converse of Theorem 52 is not necessary, since a totally strong self-complement BFGS , as shown in Figure 10, is strong and has a totally strong self-complement underlying graph structure, but and are not constant valued functions.

4. Conclusions

Graph-theoretical concepts are widely used to study and model various applications in different areas. However, in many cases, some aspects of a graph-theoretical problem may be vague or uncertain. It is natural to deal with the vagueness and uncertainty using the methods of fuzzy sets. Since bipolar fuzzy set has shown advantages in handling vagueness and uncertainty than fuzzy set, we have applied the concept of bipolar fuzzy sets to graph structures. We have introduced the concept of bipolar fuzzy graph structures. We are extending our work to (1) bipolar fuzzy soft graph structures, (2) soft graph structures, (3) rough fuzzy soft graph structures, and (4) roughness in fuzzy graph structures.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are thankful to the anonymous referees for the critical review of their paper.

References

E. Sampathkumar, “Generalized graph structures,” Bulletin of Kerala Mathematics Association, vol. 3, no. 2, pp. 65–123, 2006.

L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.

W.-R. Zhang, “(Yin) (Yang) bipolar fuzzy sets,” in Proceedings of the IEEE International Conference on Fuzzy Systems Proceedings and the IEEE World Congress on Computational Intelligence (FUZZ-IEEE '98), vol. 1, pp. 835–840, IEEE, Anchorage, Alaska, USA, May 1998.

A. Kauffman, Introduction a la Theorie des Sous-Emsembles Flous, vol. 1, Masson et Cie, 1973.

A. Rosenfeld, “Fuzzy graphs,” in Fuzzy Sets and Their Applications, L. A. Zadeh, K. S. Fu, and M. Shimura, Eds., pp. 77–95, Academic Press, New York, NY, USA, 1975.

P. Bhattacharya, “Some remarks on fuzzy graphs,” Pattern Recognition Letters, vol. 6, no. 5, pp. 297–302, 1987.

J. N. Mordeson, P. S. Nair, and J. N. Mordeson, Fuzzy Graphs and Fuzzy Hypergraphs, Second Edition 2001, Physica, Heidelberg, Germany, 2nd edition, 1998.

T. Dinesh, A study on graph structures, incidence algebras and their fuzzy analogues [Ph.D. thesis], Kannur University, Kannur, India, 2011.

M. Akram, “Bipolar fuzzy graphs,” Information Sciences, vol. 181, no. 24, pp. 5548–5564, 2011.

M. Akram, “Bipolar fuzzy graphs with applications,” Knowledge-Based Systems, vol. 39, pp. 1–8, 2013.

M. Akram and W. A. Dudek, “Regular bipolar fuzzy graphs,” Neural Computing & Applications, vol. 21, no. 1, pp. 197–205, 2012.

M. Akram, S.-G. Li, and K. P. Shum, “Antipodal bipolar fuzzy graphs,” Italian Journal of Pure and Applied Mathematics, vol. 31, pp. 97–110, 2013.

M. Akram, W. A. Dudek, and S. Sarwar, “Properties of bipolar fuzzy hypergraphs,” Italian Journal of Pure and Applied Mathematics, vol. 31, pp. 426–458, 2013.

N. O. Al-Shehrie and M. Akram, “Bipolar fuzzy competition graphs,” Ars Combinatoria, vol. 121, pp. 385–402, 2015.

K.-M. Lee, “Bipolar-valued fuzzy sets and their basic operations,” in Proceedings of the International Conference, pp. 307–317, Bangkok, Thailand, 2000.

S. Mathew and M. S. Sunitha, “Node connectivity and arc connectivity of a fuzzy graph,” Information Sciences, vol. 180, no. 4, pp. 519–531, 2010.

L. A. Zadeh, “Similarity relations and fuzzy orderings,” Information Sciences, vol. 3, no. 2, pp. 177–200, 1971.

W.-R. Zhang, “Bipolar fuzzy sets and relations: a computational framework forcognitive modeling and multiagent decision analysis,” in Proceedings of the Industrial Fuzzy Control and Intelligent Systems Conference, and the NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, Fuzzy Information Processing Society Biannual Conference, pp. 305–309, San Antonio, Tex, USA, December 2014.