Application of Bipolar Fuzzy Sets in Graph Structures

38 Pages·2016·0.23 MB·English

by  Akram M.

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
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.