# Application of Bipolar Fuzzy Sets in Graph Structures

Authors: Muhammad Akram; Rabia Akmal
Journal: Applied Computational Intelligence and Soft Computing
 (2016)

---

## 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 fuzzyN
i-cycle, bipolar fuzzy N
i-tree, bipolar fuzzy N
i-cut vertex, and bipolar fuzzy N
i-bridge, and illustrate these notions by several examples. We study ϕ-complement, self-complement, strong self-complement, and totally strong self-complement in bipolar fuzzy graph structures, and we investigate some of their interesting properties.


---

## Body

## 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 [
-
1,1
]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (
0,1
] of an element indicates that the element somewhat satisfies the property, and the membership degree [
-
1,0
) 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 N
i-cycle, bipolar fuzzy N
i-tree, bipolar fuzzy N
i-cut vertex, and bipolar fuzzy N
i-bridge and illustrate these notions by several examples. We present ϕ-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.Agraph structure
G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
k
) consists of a nonempty set U together with relations E
1
,
E
2
,
…
,
E
k on U, which are mutually disjoint such that each E
i is irreflexive and symmetric. If (
u
,
v
)
∈
E
i for some i
,
1
≤
i
≤
k, we call it an E
i-edge and write it as “u
v.” Agraph structure
G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
k
) iscomplete, if (i) each edge E
i
,
1
≤
i
≤
k, appears at least once in G
∗; (ii) between each pair of vertices u
v in U, u
v is an E
i-edge for some i
,
1
≤
i
≤
k
. Agraph structure
G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
k
) isconnected, if the underlying graph isconnected. In agraph structure, E
i-path between two vertices u and v, is the path which consists of only E
i-edges for some i, and similarly, E
i-cycle is the cycle which consists of only E
i-edges for some i. Agraph structure is atree if it is connected and contains no cycle or equivalently the underlying graph is a tree. G
∗ is an E
i-tree, if thesubgraph structure induced by E
i-edges is atree. Similarly, G
∗ is an E
1
E
2
⋯
E
j-tree, if G
∗ is an E
i-tree for each j
,
1
≤
j
≤
k
. Agraph structure is an E
i-forest, if thesubgraph structure induced by E
i-edges is aforest, that is, if it has no E
i-cycles. Let S
⊆
U; then thesubgraph structure
〈
S
〉 induced by S has vertex set S, where two vertices u and v in 〈
S
〉 are joined by an E
i-edge, 1
≤
i
≤
k, if and only if, they are joined by an E
i-edge in G
∗. For some i
,
1
≤
i
≤
k, the E
i-subgraph induced by S is denoted by E
i-〈
S
〉
. It has only those E
i-edges of G
∗, joining vertices in S. If T is a subset of edge set in G
∗, then subgraph structure 〈
T
〉 induced by T has the vertex set, “the end vertices in T”, whose edges are those in T
. Let G
∗
=
(
U
1
,
E
1
,
E
2
,
…
,
E
m
) and H
∗
=
(
U
2
,
E
1
′
,
E
2
′
,
…
,
E
n
′
) begraph structures. Then G
∗ and H
∗ areisomorphic, if (i) m
=
n, (ii) there exist a bijection f
:
U
1
→
U
2 and a bijection ϕ
:
{
E
1
,
E
2
,
…
,
E
n
}
→
{
E
1
′
,
E
2
′
,
…
,
E
n
′
}, say E
i
→
E
j
′, 1
≤
i
,
j
≤
n, such that for all u
,
v
∈
U
1, u
v
∈
E
i implies that f
(
u
)
f
(
v
)
∈
E
j
′.Twograph structures
G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
k
) and H
∗
=
(
U
,
E
1
′
,
E
2
′
,
…
,
E
k
′
), on the same vertex set U, areidentical, if there exists a bijection f
:
U
→
U, such that for all u and v in U
,
u
v is an E
i-edge in G
∗, then f
(
u
)
f
(
v
) is an E
i
′-edge in H
∗, where 1
≤
i
≤
k and E
i
≃
E
i
′
∀
i. Let ϕ be a permutation on {
E
1
,
E
2
,
…
,
E
k
}. Then the ϕ-cyclic complement of G
∗, denoted by (
G
∗
)
ϕ
c, is obtained by replacing E
i by ϕ
(
E
i), 1
≤
i
≤
k. Let G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
k
) be agraph structure and ϕ a permutation on {
E
1
,
E
2
,
…
,
E
k
}; then(i)
G
∗ is ϕ-self complementary, if G
∗ isisomorphic to (
G
∗
)
ϕ
c; the ϕ-cyclic complement of G
∗ and G
∗ isself-complement, if ϕ
≠ identity permutation.
(ii)
G
∗ isstrong
ϕ-self complementary, if G
∗ isidentical to (
G
∗
)
ϕ
c; the ϕ-complement of G
∗ and G
∗ isstrong self-complement, if ϕ
≠ identity permutation.Definition 1 (see [2]).
Afuzzy subset
μ on a set X is a map μ
:
X
→
[
0,1
]. Afuzzy binary relation on X is a fuzzy subset μ on X
×
X. By a fuzzy relation we mean a fuzzy binary relation given by μ
:
X
×
X
→
[
0,1
].Definition 2 (see [8]).
LetG
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
k
) be a graph structure and let ν
,
ρ
1
,
ρ
2
,
…
,
ρ
k be the fuzzy subsets of U
,
E
1
,
E
2
,
…
,
E
k, respectively, such that(1)
0
≤
ρ
i
x
y
≤
μ
x
∧
μ
y
∀
x
,
y
∈
U
,
i
=
1,2
,
…
,
k
.Then G
=
(
ν
,
ρ
1
,
ρ
2
,
…
,
ρ
k
) is a fuzzy graph structure of G
∗.Definition 3 (see [8]).
LetG
=
(
ν
,
ρ
1
,
ρ
2
,
…
,
ρ
k
) be a fuzzy graph structure of a graph structure G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
k
)
. Then F
=
(
ν
,
τ
1
,
τ
2
,
…
,
τ
k
) is a partial fuzzy spanning subgraph structure of G if τ
i
⊆
ρ
i for i
=
1,2
,
…
,
k
.Definition 4 (see [8]).
LetG
∗ be a graph structure and let G be a fuzzy graph structure of G
∗. If x
y
∈
supp
⁡
(
ρ
i
), then “x
y” is said to be a ρ
i-edge of G.Definition 5 (see [8]).
The strength of aρ
i-path x
0
x
1
⋯
x
n of a fuzzy graph structure G is ⋀
j
=
1
n
ρ
i
(
x
j
-
1
x
j
) for i
=
1,2
,
…
,
k.Definition 6 (see [8]).
In a fuzzy graph structureG
,
ρ
i
2
(
x
y
)
=
ρ
i
∘
ρ
i
(
x
y
)
=
⋁
z
{
ρ
i
(
x
z
)
∧
ρ
i
(
z
y
)
}, ρ
i
j
(
x
y
)
=
(
ρ
i
j
-
1
∘
ρ
i
)
(
x
y
)
=
⋁
z
{
ρ
i
j
-
1
(
x
z
)
∧
ρ
i
(
z
y
)
}, j
=
2,3
,
…
,
m, for any m
≥
2
. Also ρ
i
∞
x
y
=
⋁
{
ρ
i
j
(
x
y
)
,
j
=
1,2
,
…
}.Definition 7 (see [8]).
Letx
y be a ρ
i-edge of G
=
(
ν
,
ρ
1
,
ρ
2
,
…
,
ρ
n
)
. Let (
ν
,
ρ
1
′
,
ρ
2
′
,
…
,
ρ
n
′
) be a partial fuzzy spanning subgraph structure obtained by deleting “x
y” with ρ
i
′
(
x
y
)
=
0 and ρ
i
′
(
x
1
y
1
)
=
ρ
i
(
x
1
y
1
)
∀
ρ
i-edges (
x
1
y
1
) other than (
x
,
y
)
. If ρ
i
∞
(
u
v
)
>
ρ
i
′
∞
(
u
v
) for some u
v
∈ supp(
ρ
i
), then x
y is a ρ
i-bridge.Definition 8 (see [8]).
LetG
′
=
(
ν
,
ρ
1
′
,
ρ
2
′
,
…
,
ρ
n
′
) be the partial fuzzy subgraph structure obtained by deleting vertex w of G, that is, ν
′
(
w
)
=
0 and ν
′
(
v
)
=
ν
(
v
)
∀
v
≠
w
,
ρ
i
′
(
v
w
)
=
0
∀
v
∈
U
r and ρ
i
′
(
u
v
)
=
ρ
i
(
u
v
)
∀
u
v
≠
w
v
,
i
=
1,2
,
…
,
k
. Then a vertex w of G is a ρ
i-cut vertex if ρ
i
∞
(
u
v
)
>
ρ
i
′
∞
(
u
v
) for some u
,
v with u
,
v
≠
w
.Definition 9 (see [8]).
G
=
(
ν
,
ρ
1
,
ρ
2
,
…
,
ρ
k
) is a ρ
i-cycle if and only if (
supp
⁡
ν
,
supp
⁡
ρ
1
,
supp
⁡
ρ
2
,
…
,
supp
⁡
(
ρ
k
)
) is a E
i-cycle.Definition 10 (see [8]).
G
=
(
ν
,
ρ
1
,
ρ
2
,
…
,
ρ
k
) is a fuzzy ρ
i-cycle if and only if (
supp
⁡
ν
,
supp
⁡
ρ
1
,
supp
⁡
ρ
2
,
…
,
supp
⁡
(
ρ
k
)
) is an E
i-cycle and there exists no unique “x
y” in supp(
ρ
i
) such that ρ
i
(
x
y
)
=
⋀
{
ρ
i
(
u
v
)
∣
u
v
∈
supp
⁡
(
ρ
i
)
}.Definition 11 (see [8]).
G
=
(
ν
,
ρ
1
,
ρ
2
,
…
,
ρ
k
) is a fuzzy ρ
i-tree if it has a partial fuzzy spanning subgraph structure, F
i
=
(
ν
,
τ
1
,
τ
2
,
…
,
τ
k
), which is a τ
i-tree where for all ρ
i-edges not in F
i
,
ρ
i
(
x
y
)
<
τ
i
∞
(
x
y
)
.Definition 12 (see [8]).
LetG
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
k
) be a graph structure and let ν
,
ρ
1
,
ρ
2
,
…
,
ρ
k be the fuzzy subsets of U
,
E
1
,
E
2
,
…
,
E
k, respectively, such that(2)
0
≤
ρ
i
x
y
≤
μ
x
∧
μ
y
∀
x
,
y
∈
V
,
i
=
1,2
,
…
,
k
.Then G
=
(
ν
,
ρ
1
,
ρ
2
,
…
,
ρ
k
) is a fuzzy graph structure of G
∗
.Definition 13 (see [3]).
LetX be a nonempty set. Abipolar fuzzy set
B in X is an object having the form (3)
B
=
x
,
μ
B
P
x
,
μ
B
N
x
∣
x
∈
X
,where μ
B
P
:
X
→
[
0
,
1
] and μ
B
N
:
X
→
[
-
1
,
0
] are mappings.We use the positive membership degreeμ
B
P
(
x
) to denote the satisfaction degree of an element x to the property corresponding to a bipolar fuzzy set B and the negative membership degree μ
B
N
(
x
) to denote the satisfaction degree of an element x to some implicit counterproperty corresponding to a bipolar fuzzy set B. If μ
B
P
(
x
)
≠
0 and μ
B
N
(
x
)
=
0, it is the situation that x is regarded as having only positive satisfaction for B. If μ
B
P
(
x
)
=
0 and μ
B
N
(
x
)
≠
0, it is the situation that x does not satisfy the property of B but somewhat satisfies the counter property of B. It is possible for an element x to be such that μ
B
P
(
x
)
≠
0 and μ
B
N
(
x
)
≠
0 when the membership function of the property overlaps that of its counterproperty over some portion of X.For the sake of simplicity, we will use the symbolB
=
(
μ
B
P
,
μ
B
N
) for the bipolar fuzzy set: (4)
B
=
x
,
μ
B
P
x
,
μ
B
N
x
∣
x
∈
X
.Definition 14 (see [3]).
LetX be a nonempty set. Then we call a mapping A
=
(
μ
A
P
,
μ
A
N
)
:
X
×
X
→
[
0,1
]
×
[
-
1,0
] abipolar fuzzy relation on X such that μ
A
P
(
x
,
y
)
∈
[
0,1
] and μ
A
N
(
x
,
y
)
∈
[
-
1,0
].Definition 15 (see [9]).
A bipolar fuzzy graphG
=
(
V
,
A
,
B
) is a nonempty set V together with a pair of functions A
=
(
μ
A
P
,
μ
A
N
)
:
V
→
[
0,1
]
×
[
-
1,0
] and B
=
(
μ
B
P
,
μ
B
N
)
:
V
×
V
→
[
0,1
]
×
[
-
1,0
] such that for all x
,
y
∈
V, (5)
μ
B
P
x
,
y
≤
m
i
n
μ
A
P
x
,
μ
A
P
y
,
μ
B
N
x
,
y
≥
m
a
x
μ
A
N
x
,
μ
A
N
y
.
Notice thatμ
B
P
(
x
,
y
)
>
0, μ
B
N
(
x
,
y
)
<
0 for (
x
,
y
)
∈
V
×
V, μ
B
P
(
x
,
y
)
=
μ
B
N
(
x
,
y
)
=
0 for (
x
,
y
)
∉
V
×
V, and B is symmetric relation.


## 3. Bipolar Fuzzy Graph Structures

Definition 16.
G
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) is called abipolar fuzzy graph structure (BFGS) of a graph structure (GS) G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
) if M
=
(
μ
M
P
,
μ
M
N
) is abipolar fuzzy set on
U and for each i
=
1,2
,
…
,
n; N
i
=
(
μ
N
i
P
,
μ
N
i
N
) is abipolar fuzzy set on
E
i such that (6)
μ
N
i
P
x
y
≤
μ
M
P
x
∧
μ
M
P
y
,
μ
N
i
N
x
y
≥
μ
M
N
x
∨
μ
M
N
y
∀
x
y
∈
E
i
⊂
U
×
U
.Note that μ
N
i
P
(
x
y
)
=
0
=
μ
N
i
N
(
x
y
) for all x
y
∈
U
×
U
-
E
i and 0
<
μ
N
i
P
(
x
y
)
≤
1, -
1
≤
μ
N
i
N
(
x
y
)
<
0
∀
x
y
∈
E
i, where U and E
i
(
i
=
1,2
,
…
,
n
) are calledunderlying vertex set andunderlying
i
-edge set of G
ˇ
b, respectively.Definition 17.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) be abipolar fuzzy graph structure of agraph structure
G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
)
. If H
ˇ
b
=
(
M
′
,
N
1
′
,
N
2
′
,
…
,
N
n
′
) is abipolar fuzzy graph structure of G
∗ such that (7)
μ
M
′
P
x
≤
μ
M
P
x
,
μ
M
′
N
x
≥
μ
M
N
x
∀
x
∈
U
,
μ
N
i
′
P
x
y
≤
μ
N
i
P
x
y
,
μ
N
i
′
N
x
y
≥
μ
N
i
N
x
y
∀
x
y
∈
E
i
,
i
=
1,2
,
…
,
n
,then H
ˇ
b is called abipolar fuzzy subgraph structure of BFGS G
ˇ
b.
BFGSH
ˇ
b
=
(
M
′
,
N
1
′
,
N
2
′
,
…
,
N
n
′
) is abipolar fuzzy induced subgraph structure of G
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
), by a subset W of U if (8)
μ
M
′
P
x
=
μ
M
P
x
,
μ
M
′
N
x
=
μ
M
N
x
∀
x
∈
W
,
μ
N
i
′
P
x
y
=
μ
N
i
P
x
y
,
μ
N
i
′
N
x
y
=
μ
N
i
N
x
y
∀
x
,
y
∈
W
,
i
=
1,2
,
…
,
n
.Similarly, BFGS H
ˇ
b is abipolar fuzzy spanning subgraph structure of G
ˇ
b if M
′
=
M and (9)
μ
N
i
′
P
≤
μ
N
i
P
,
μ
N
i
′
N
≥
μ
N
i
N
,
i
=
1,2
,
…
,
n
.Example 18.
Consider a graph structureG
∗
=
(
U
,
E
1
,
E
2
) such that U
=
{
a
1
,
a
2
,
a
3
,
a
4
}
,
E
1
=
{
a
1
a
2
,
a
2
a
4
}, and E
2
=
{
a
3
a
4
,
a
1
a
4
}
.
(i) LetM
,
N
1, and N
2 be bipolar fuzzy subsets of U
,
E
1, and E
2, respectively, such that (10)
M
=
a
1
,
0.5
,
-
0.2
,
a
2
,
0.7
,
-
0.3
,
a
3
,
0.4
,
-
0.3
,
a
4
,
0.7
,
-
0.3
,
N
1
=
a
1
a
2
,
0.5
,
-
0.2
,
a
2
a
4
,
0.7
,
-
0.3
,
N
2
=
a
3
a
4
,
0.3
,
-
0.2
,
a
1
a
4
,
0.3
,
-
0.1
.Then, by direct calculations, it is easy to see that G
ˇ
b
=
(
M
,
N
1
,
N
2
) is a BFGS of G
∗ as shown in Figure 1.
(ii) ConsiderM
1
=
{
(
a
1
,
0.4
,
-
0.1
)
,
(
a
2
,
0.5
,
-
0.3
)
,
(
a
3
,
0.4
,
-
0.2
)
,
(
a
4
,
0.1
,
-
0.3
)
}, N
11
=
{
(
a
1
a
2
,
0.4
,
-
0.1
)
,
(
a
2
a
4
,
0.1
,
-
0.2
)
}, and N
12
=
{
(
a
3
a
4
,
0.1
,
-
0.2
)
,
(
a
1
a
4
,
0.1
,
-
0.0
)
}. Then, by routine calculations, it is easy to see that K
ˇ
b
=
(
M
1
,
N
11
,
N
12
) is the bipolar fuzzy subgraph structure of G
ˇ
b as shown in Figure 2.Figure 1
G
ˇ
b
=
(
M
,
N
1
,
N
2
).Figure 2
Bipolar fuzzy subgraph structure
K
ˇ
b
=
(
M
1
,
N
11
,
N
12
).Definition 19.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) be abipolar fuzzy graph structure of a graph structure G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
)
. Then x
y
∈
E
i is called abipolar fuzzy
N
i-edge or simply N
i-edge, if(11)
μ
N
i
P
x
y
>
0
or
μ
N
i
N
x
y
<
0
.Thensupport of N
i, i
=
1,2
,
…
,
n, consequently, is (12)
supp
⁡
N
i
=
x
y
∈
E
i
:
μ
N
i
P
x
y
>
0
,
μ
N
i
N
x
y
<
0
.Definition 20.
N
i-path in a BFGS G
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) of a graph structure G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
) is a sequence a
1
,
a
2
,
…
,
a
m of distinct vertices (except the choice a
m
=
a
1)in U, such that a
j
-
1
a
j is abipolar fuzzy
N
i-edge for all j
=
2,3
,
…
,
m
.Definition 21.
A BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) with underlying vertex set U is said to be N
i-strong for some i
∈
{
1,2
,
3
,
…
,
n
} if for all x
y
∈
supp
⁡
(
N
i
)
(13)
μ
N
i
P
x
y
=
μ
M
P
x
∧
μ
M
P
y
,
μ
N
i
N
x
y
=
μ
M
N
x
∨
μ
M
N
y
.A BFGS G
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) is said to bestrong if it is N
i
-strong BFGS for all i
∈
{
1,2
,
3
,
…
,
n
}
.Example 22.
Consider BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
) as shown in Figure 3.
ThenG
ˇ
b is a strong BFGS since it is both N
1- and N
2-strong.Figure 3
BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
).Definition 23.
A BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) with underlying vertex set U is said to becomplete or N
1
N
2
⋯
N
n-complete, if the following are true: (i)
G
ˇ
b a isstrong BFGS.
(ii)
supp
⁡
(
N
i
)
≠
∅
∀
i
=
1,2
,
3
,
…
,
n.
(iii)
For each pair of verticesx
,
y
∈
U
,
x
y is an N
i-edge for some i.Example 24.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
) be BFGS of graph structure G
∗
=
(
U
,
E
1
,
E
2
) such that U
=
{
a
1
,
a
2
,
a
3
}
,
E
1
=
{
a
2
a
3
}, and E
2
=
{
a
1
a
2
,
a
1
a
3
} as shown in Figure 4. By routine calculations, it is easy to see that G
ˇ
b is a strong BFGS.
Moreover,supp
⁡
(
N
1
)
≠
∅
,
supp
⁡
(
N
2
)
≠
∅, and every pair of vertices belonging to U is either an N
1-edge or an N
2-edge. So G
ˇ
b is a complete BFGS, that is, N
1
N
2-complete BFGS.Figure 4
G
ˇ
b
=
(
M
,
N
1
,
N
2
).Definition 25.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) be a BFGS with underlying vertex set U
. Then positive and negative strengths of a N
i-path “P
N
i
=
a
1
a
2
⋯
a
m” are called gain and loss of that N
i-path and denoted by G
.
P
N
i and L
.
P
N
i, respectively, such that (14)
G
.
P
N
i
=
⋀
j
=
2
m
μ
N
i
P
a
j
-
1
a
j
,
L
.
P
N
i
=
⋁
j
=
2
m
μ
N
i
N
a
j
-
1
a
j
.Example 26.
Consider a BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
) as shown in Figure 4. We note that P
N
2
=
a
1
a
3
a
4
a
1 is an N
2-path. So G
.
P
N
2
=
μ
N
2
P
(
a
3
a
1
)
∧
μ
N
2
P
(
a
1
a
2
)
=
0.5
∧
0.4
=
0.4. Consider (15)
L
.
P
N
2
=
μ
N
2
N
a
3
a
1
∨
μ
N
2
N
a
1
a
2
=
-
0.4
∨
-
0.4
=
-
0.4
=
0.4
.Definition 27.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) be a BFGS withunderlying vertex set
U
. Then(i)
N
i-gain of connectedness between x and y is defined by μ
N
i
∞
,
+
(
x
y
)
=
⋁
j
≥
1
{
μ
N
i
j
,
+
(
x
y
)
}, such that μ
N
i
j
,
+
(
x
y
)
=
(
μ
N
i
j
-
1
,
+
∘
μ
N
i
1
,
+
)
(
x
y
) for j
≥
2 and μ
N
i
2
,
+
(
x
y
)
=
(
μ
N
i
1
,
+
∘
μ
N
i
1
,
+
)
(
x
y
)
=
⋁
z
{
μ
N
i
1
,
+
(
x
z
)
∧
μ
N
i
1
,
+
(
z
y
)
}, where μ
N
i
1
,
+
=
μ
N
i
P
,
∀
i
.
(ii)
N
i-loss of connectedness between x and y is defined by μ
N
i
∞
,
-
(
x
y
)
=
⋁
j
≥
1
{
μ
N
i
j
,
-
(
x
y
)
}, such that μ
N
i
j
,
-
(
x
y
)
=
(
μ
N
i
j
-
1
,
-
∘
μ
N
i
1
,
-
)
(
x
y
) for j
≥
2 and μ
N
i
2
,
-
(
x
y
)
=
(
μ
N
i
1
,
-
∘
μ
N
i
1
,
-
)
(
x
y
)
=
⋁
z
{
μ
N
i
1
,
-
(
x
z
)
∧
μ
N
i
1
,
-
(
z
y
)
}, where μ
N
i
1
,
-
=
μ
N
i
N
,
∀
i
.Example 28.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
) be BFGS of graph structure G
=
(
U
,
E
1
,
E
2
) such that U
=
{
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
}, E
1
=
{
a
1
a
6
,
a
2
a
3
,
a
2
a
5
,
a
3
a
4
,
a
4
a
5
}, and E
2
=
{
a
1
a
3
,
a
1
a
2
,
a
4
a
6
,
a
5
a
6
}, as is shown in Figure 5.
Sinceμ
N
1
1
,
+
(
a
2
a
3
)
=
0.3, μ
N
1
1
,
+
(
a
2
a
4
)
=
0.0, μ
N
1
1
,
+
(
a
2
a
5
)
=
0.4, μ
N
1
1
,
+
(
a
3
a
4
)
=
0.5, μ
N
1
1
,
+
(
a
5
a
3
)
=
0.0, μ
N
1
1
,
+
(
a
4
a
5
)
=
0.3, and μ
N
1
1
,
+
(
a
1
a
6
)
=
0.3, therefore(16)
μ
N
1
2
,
+
a
2
a
3
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
2
a
3
=
μ
N
1
1
,
+
a
2
a
4
∧
μ
N
1
1
,
+
a
4
a
3
∨
μ
N
1
1
,
+
a
2
a
5
∧
μ
N
1
1
,
+
a
5
a
3
=
0.0
∧
0.5
∨
0.4
∧
0.0
=
0
,
μ
N
1
2
,
+
a
2
a
4
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
2
a
4
=
μ
N
1
2
,
+
a
2
a
3
∧
μ
N
1
1
,
+
a
3
a
4
∨
μ
N
1
1
,
+
a
2
a
5
∧
μ
N
1
1
,
+
a
5
a
4
=
0.3
∧
0.5
∨
0.4
∧
0.3
=
0.3
,
μ
N
1
2
,
+
a
2
a
5
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
2
a
5
=
μ
N
1
1
,
+
a
2
a
3
∧
μ
N
1
1
,
+
a
3
a
5
∨
μ
N
1
1
,
+
a
2
a
4
∧
μ
N
1
1
,
+
a
4
a
5
=
0.3
∧
0.0
∨
0.0
∧
0.3
=
0
,
μ
N
1
2
,
+
a
3
a
4
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
3
a
4
=
μ
N
1
1
,
+
a
3
a
2
∧
μ
N
1
1
,
+
a
2
a
4
∨
μ
N
1
1
,
+
a
3
a
5
∧
μ
N
1
1
,
+
a
5
a
4
=
0.3
∧
0.0
∨
0.0
∧
0.3
=
0
,
μ
N
1
2
,
+
a
3
a
5
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
3
a
5
=
μ
N
1
1
,
+
a
3
a
2
∧
μ
N
1
1
,
+
a
2
a
5
∨
μ
N
1
1
,
+
a
3
a
4
∧
μ
N
1
1
,
+
a
4
a
5
=
0.3
∧
0.4
∨
0.5
∧
0.3
=
0.3
,
μ
N
1
2
,
+
a
4
a
5
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
4
a
5
=
μ
N
1
1
,
+
a
4
a
2
∧
μ
N
1
1
,
+
a
2
a
5
∨
μ
N
1
1
,
+
a
4
a
3
∧
μ
N
1
1
,
+
a
3
a
5
=
0.0
∧
0.4
∨
0.5
∧
0.0
=
0
,
μ
N
1
2
,
+
a
1
a
6
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
1
a
6
=
0
,
μ
N
1
3
,
+
a
2
a
3
=
μ
2
,
+
n
N
1
∘
μ
N
1
1
,
+
a
2
a
3
=
μ
N
1
2
,
+
a
2
a
4
∧
μ
N
1
1
,
+
a
4
a
3
∨
μ
N
1
2
,
+
a
2
a
5
∧
μ
N
1
1
,
+
a
5
a
3
=
0.3
∧
0.5
∨
0.0
∧
0.0
=
0.3
,
μ
N
1
3
,
+
a
2
a
4
=
μ
2
,
+
n
N
1
∘
μ
N
1
1
,
+
a
2
a
4
=
μ
N
1
2
,
+
a
2
a
3
∧
μ
N
1
1
,
+
a
3
a
4
∨
μ
N
1
2
,
+
a
2
a
5
∧
μ
N
1
1
,
+
a
5
a
4
=
0.0
∧
0.5
∨
0.0
∧
0.3
=
0.0
,
μ
N
1
3
,
+
a
2
a
5
=
μ
2
,
+
n
N
1
∘
μ
N
1
1
,
+
a
2
a
5
=
μ
N
1
2
,
+
a
2
a
3
∧
μ
N
1
1
,
+
a
3
a
5
∨
μ
N
1
2
,
+
a
2
a
4
∧
μ
N
1
1
,
+
a
4
a
5
=
0.0
∧
0.0
∨
0.3
∧
0.3
=
0.3
,
μ
N
1
3
,
+
a
3
a
4
=
μ
2
,
+
n
N
1
∘
μ
N
1
1
,
+
a
3
a
4
=
μ
N
1
2
,
+
a
3
a
2
∧
μ
N
1
1
,
+
a
2
a
4
∨
μ
N
1
2
,
+
a
3
a
5
∧
μ
N
1
1
,
+
a
5
a
4
=
0.0
∧
0.0
∨
0.3
∧
0.3
=
0.3
,
μ
N
1
3
,
+
a
3
a
5
=
μ
2
,
+
n
N
1
∘
μ
N
1
1
,
+
a
3
a
5
=
μ
N
1
2
,
+
a
3
a
2
∧
μ
N
1
1
,
+
a
2
a
5
∨
μ
N
1
2
,
+
a
3
a
4
∧
μ
N
1
1
,
+
a
4
a
5
=
0.0
∧
0.4
∨
0.0
∧
0.3
=
0.0
,
μ
N
1
3
,
+
a
4
a
5
=
μ
2
,
+
n
N
1
∘
μ
N
1
1
,
+
a
4
a
5
=
μ
N
1
2
,
+
a
4
a
2
∧
μ
N
1
1
,
+
a
2
a
5
∨
μ
N
1
2
,
+
a
4
a
3
∧
μ
N
1
1
,
+
a
3
a
5
=
0.3
∧
0.4
∨
0.0
∧
0.0
=
0.3
,
μ
N
1
3
,
+
a
1
a
6
=
μ
2
,
+
n
N
1
∘
μ
N
1
1
,
+
a
1
a
6
=
0
.
Similarly,(17)
μ
N
1
4
,
+
a
2
a
3
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
2
a
3
=
0
,
μ
N
1
4
,
+
a
2
a
4
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
2
a
4
=
0.3
,
μ
N
1
4
,
+
a
2
a
5
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
2
a
5
=
0
,
μ
N
1
4
,
+
a
3
a
4
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
3
a
4
=
0
,
μ
N
1
4
,
+
a
3
a
5
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
3
a
5
=
0.3
,
μ
N
1
4
,
+
a
4
a
5
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
4
a
5
=
0
,
μ
N
1
4
,
+
a
1
a
6
=
μ
1
,
+
n
N
1
∘
μ
N
1
1
,
+
a
1
a
6
=
0
.This implies that(18)
μ
N
1
∞
,
+
a
2
a
3
=
∨
0.3,0.0,0.3,0.0
=
0.3
,
μ
N
1
∞
,
+
a
2
a
4
=
∨
0.0,0.3,0.0,0.3
=
0.3
,
μ
N
1
∞
,
+
a
2
a
5
=
∨
0.4,0.0,0.3,0.0
=
0.4
,
μ
N
1
∞
,
+
a
3
a
4
=
∨
0.5,0.0,0.3,0.0
=
0.5
,
μ
N
1
∞
,
+
a
3
a
5
=
∨
0.0,0.3,0.0,0.3
=
0.3
,
μ
N
1
∞
,
+
a
4
a
5
=
∨
0.3,0.0,0.3,0.0
=
0.3
,
μ
N
1
∞
,
+
a
1
a
6
=
∨
0.3,0.0,0.0,0.0
=
0.3
.Since(19)
μ
N
1
1
,
-
a
2
a
3
=
0.5
,
μ
N
1
1
,
-
a
2
a
4
=
0.0
,
μ
N
1
1
,
-
a
2
a
5
=
0.4
,
μ
N
1
1
,
-
a
3
a
4
=
0.7
,
μ
N
1
1
,
-
a
5
a
3
=
0.0
,
μ
N
1
1
,
-
a
4
a
5
=
0.3
,
μ
N
1
1
,
-
a
1
a
6
=
0.2
,we have(20)
μ
N
1
2
,
-
a
2
a
3
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
2
a
3
=
μ
N
1
1
,
-
a
2
a
4
∧
μ
N
1
1
,
-
a
4
a
3
∨
μ
N
1
1
,
-
a
2
a
5
∧
μ
N
1
1
,
-
a
5
a
3
=
0.0
∧
0.7
∨
0.4
∧
0.0
=
0.0
,
μ
N
1
2
,
-
a
2
a
4
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
2
a
4
=
μ
N
1
2
,
-
a
2
a
3
∧
μ
N
1
1
,
-
a
3
a
4
∨
μ
N
1
1
,
-
a
2
a
5
∧
μ
N
1
1
,
-
a
5
a
4
=
0.5
∧
0.7
∨
0.4
∧
0.3
=
0.5
,
μ
N
1
2
,
-
a
2
a
5
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
2
a
5
=
μ
N
1
1
,
-
a
2
a
3
∧
μ
N
1
1
,
-
a
3
a
5
∨
μ
N
1
1
,
-
a
2
a
4
∧
μ
N
1
1
,
-
a
4
a
5
=
0.5
∧
0.0
∨
0.0
∧
0.3
=
0.0
,
μ
N
1
2
,
-
a
3
a
4
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
3
a
4
=
μ
N
1
1
,
-
a
3
a
2
∧
μ
N
1
1
,
-
a
2
a
4
∨
μ
N
1
1
,
-
a
3
a
5
∧
μ
N
1
1
,
-
a
5
a
4
=
0.5
∧
0.0
∨
0.0
∧
0.3
=
0.0
,
μ
N
1
2
,
-
a
3
a
5
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
3
a
5
=
μ
N
1
1
,
-
a
3
a
2
∧
μ
N
1
1
,
-
a
2
a
5
∨
μ
N
1
1
,
-
a
3
a
4
∧
μ
N
1
1
,
-
a
4
a
5
=
0.5
∧
0.4
∨
0.7
∧
0.3
=
0.4
,
μ
N
1
2
,
-
a
4
a
5
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
4
a
5
=
μ
N
1
1
,
-
a
4
a
2
∧
μ
N
1
1
,
-
a
2
a
5
∨
μ
N
1
1
,
-
a
4
a
3
∧
μ
N
1
1
,
-
a
3
a
5
=
0.0
∧
0.4
∨
0.7
∧
0.0
=
0.0
,
μ
N
1
2
,
-
a
1
a
6
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
1
a
6
=
0
,
μ
N
1
3
,
-
a
2
a
3
=
μ
2
,
-
n
N
1
∘
μ
N
1
1
,
-
a
2
a
3
=
μ
N
1
2
,
-
a
2
a
4
∧
μ
N
1
1
,
-
a
4
a
3
∨
μ
N
1
2
,
-
a
2
a
5
∧
μ
N
1
1
,
-
a
5
a
3
=
0.5
∧
0.7
∨
0.0
∧
0.0
=
0.5
,
μ
N
1
3
,
-
a
2
a
4
=
μ
2
,
-
n
N
1
∘
μ
N
1
1
,
-
a
2
a
4
=
μ
N
1
2
,
-
a
2
a
3
∧
μ
N
1
1
,
-
a
3
a
4
∨
μ
N
1
2
,
-
a
2
a
5
∧
μ
N
1
1
,
-
a
5
a
4
=
0.0
∧
0.7
∨
0.0
∧
0.3
=
0.0
,
μ
N
1
3
,
-
a
2
a
5
=
μ
2
,
-
n
N
1
∘
μ
N
1
1
,
-
a
2
a
5
=
μ
N
1
2
,
-
a
2
a
3
∧
μ
N
1
1
,
-
a
3
a
5
∨
μ
N
1
2
,
-
a
2
a
4
∧
μ
N
1
1
,
-
a
4
a
5
=
0.0
∧
0.0
∨
0.5
∧
0.3
=
0.3
,
μ
N
1
3
,
-
a
3
a
4
=
μ
2
,
-
n
N
1
∘
μ
N
1
1
,
-
a
3
a
4
=
μ
N
1
2
,
-
a
3
a
2
∧
μ
N
1
1
,
-
a
2
a
4
∨
μ
N
1
2
,
-
a
3
a
5
∧
μ
N
1
1
,
-
a
5
a
4
=
0.0
∧
0.0
∨
0.4
∧
0.3
=
0.3
,
μ
N
1
3
,
-
a
3
a
5
=
μ
2
,
-
n
N
1
∘
μ
N
1
1
,
-
a
3
a
5
=
μ
N
1
2
,
-
a
3
a
2
∧
μ
N
1
1
,
-
a
2
a
5
∨
μ
N
1
2
,
-
a
3
a
4
∧
μ
N
1
1
,
-
a
4
a
5
=
0.0
∧
0.4
∨
0.0
∧
0.3
=
0.0
,
μ
N
1
3
,
-
a
4
a
5
=
μ
2
,
-
n
N
1
∘
μ
N
1
1
,
-
a
4
a
5
=
μ
N
1
2
,
-
a
4
a
2
∧
μ
N
1
1
,
-
a
2
a
5
∨
μ
N
1
2
,
-
a
4
a
3
∧
μ
N
1
1
,
-
a
3
a
5
=
0.5
∧
0.4
∨
0.0
∧
0.0
=
0.4
,
μ
N
1
3
,
-
a
1
a
6
=
μ
2
,
-
n
N
1
∘
μ
N
1
1
,
-
a
1
a
6
=
0
.Similarly,(21)
μ
N
1
4
,
-
a
2
a
3
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
2
a
3
=
0
,
μ
N
1
4
,
-
a
2
a
4
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
2
a
4
=
0.5
,
μ
N
1
4
,
-
a
2
a
5
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
2
a
5
=
0
,
μ
N
1
4
,
-
a
3
a
4
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
3
a
4
=
0
,
μ
N
1
4
,
-
a
3
a
5
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
3
a
5
=
0.4
,
μ
N
1
4
,
-
a
4
a
5
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
4
a
5
=
0
,
μ
N
1
4
,
-
a
1
a
6
=
μ
1
,
-
n
N
1
∘
μ
N
1
1
,
-
a
1
a
6
=
0
.This implies that(22)
μ
N
1
∞
,
-
a
2
a
3
=
∨
0.5,0.0,0.5,0.0
=
0.5
,
μ
N
1
∞
,
-
a
2
a
4
=
∨
0.0,0.5,0.0,0.5
=
0.5
,
μ
N
1
∞
,
-
a
2
a
5
=
∨
0.4,0.0,0.3,0.0
=
0.4
,
μ
N
1
∞
,
-
a
3
a
4
=
∨
0.7,0.0,0.3,0.0
=
0.7
,
μ
N
1
∞
,
-
a
3
a
5
=
∨
0.0,0.4,0.0,0.4
=
0.4
,
μ
N
1
∞
,
-
a
4
a
5
=
∨
0.3,0.0,0.4,0.0
=
0.4
,
μ
N
1
∞
,
-
a
1
a
6
=
∨
0.3,0.0
,
0.0,0.0
=
0.2
.For all the remaining pairs of vertices, N
1-loss and N
1-gain of connectedness are zero.Figure 5
G
ˇ
b
=
(
M
,
N
1
,
N
2
).Definition 29.
A BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) of agraph structure
G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
) is an N
i-cycle if (
supp
⁡
M
,
supp
⁡
N
1
,
supp
⁡
N
2
,
…
,
supp
⁡
(
N
n
)
) is an E
i-cycle.Definition 30.
A BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) of agraph structure
G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
) is abipolar fuzzy
N
i-cycle for some i if (i)
G
ˇ
b is an N
i-cycle;
(ii)
there is no uniqueN
i-edge
u
v in G
ˇ
b such that μ
N
i
P
(
u
v
)
=
min
⁡
{
μ
N
i
P
(
x
y
)
:
x
y
∈
E
i
=
supp
⁡
(
N
i
)
} or μ
N
i
N
(
u
v
)
=
max
⁡
{
μ
N
i
N
(
x
y
)
:
x
y
∈
E
i
=
supp
⁡
(
N
i
)
}.Example 31.
Consider BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
) as shown in Figure 3. Then G
ˇ
b is an N
1-cycle as well as bipolar fuzzy N
1-cycle, since (
supp
⁡
M
,
supp
⁡
N
1
,
supp
⁡
(
N
2
)
) is an E
1-cycle and there are two N
1-edges with minimum positive degree and more than one N
1-edge with maximum negative degree of all N
1-edges.Definition 32.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) be a BFGS of a graph structure G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
) and x a vertex of G
ˇ
b. Let (
M
′
,
N
1
′
,
N
2
′
,
…
,
N
n
′
) be abipolar fuzy subgraph structure of G
ˇ
b induced by U
∖
{
x
} such that (23)
μ
M
′
P
x
=
0
=
μ
M
′
N
x
,
μ
N
i
′
P
x
v
=
0
=
μ
N
i
′
N
x
v
∀
e
d
g
e
s
x
v
∈
G
ˇ
b
,
μ
M
′
P
v
=
μ
M
P
v
,
μ
M
′
N
v
=
μ
M
N
v
,
∀
v
≠
x
,
μ
N
i
′
P
u
v
=
μ
N
i
P
u
v
,
μ
N
i
′
N
u
v
=
μ
N
i
N
u
v
∀
i
,
such  that
u
≠
x
,
v
≠
x
.Then x is abipolar fuzzy
N
i-cut vertex for some i, if(24)
μ
N
i
∞
,
+
u
v
>
μ
N
i
′
∞
,
+
u
v
,
μ
N
i
∞
,
-
u
v
>
μ
N
i
′
∞
,
-
u
v
for  some
u
,
v
∈
U
∖
x
.And, x is an N
i-P bipolar fuzzy cut vertex if only the first condition holds and a N
i-N bipolar fuzzy cut vertex if only the second condition holds.Example 33.
Consider BFSGG
ˇ
b
=
(
M
,
N
1
,
N
2
) as considered in Example 28 and shown in Figure 5; after deleting vertex a
2, the resulting bipolar fuzzy subgraph structure will be as shown in Figure 6.
Thena
2 is a bipolar fuzzy N
1-N cut vertex since(25)
μ
N
1
∞
,
-
a
3
a
4
=
0.7
=
μ
N
1
′
∞
,
-
a
3
a
4
,
μ
N
1
∞
,
-
a
3
a
5
=
0.4
>
0.3
=
μ
N
1
∞
,
-
a
3
a
5
,
μ
N
1
∞
,
-
a
4
a
5
=
0.4
>
0.3
=
μ
N
1
′
∞
,
-
a
4
a
5
,
μ
N
1
∞
,
-
a
1
a
6
=
0.2
=
μ
N
1
∞
,
-
a
1
a
6
.Figure 6
Bipolar fuzzy subgraph structure
(
M
∖
{
a
2
}
,
N
1
′
,
N
2
′
).Definition 34.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) be a BFGS of a graph structure G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
) and let x
y be an N
i-edge. Let (
M
,
N
1
′
,
N
2
′
,
…
,
N
n
′
) be abipolar fuzzy spanning subgraph structure of G
ˇ
b, obtained by taking (26)
μ
N
i
′
P
x
y
=
0
=
μ
N
i
′
N
x
y
,
μ
N
i
′
P
u
v
=
μ
N
i
P
u
v
,
μ
N
i
′
N
u
v
=
μ
N
i
N
u
v
∀
e
d
g
e
s
u
v
≠
x
y
.Then x
y is abipolar fuzzy
N
i-bridge if (27)
μ
N
i
∞
,
+
u
v
>
μ
N
i
′
∞
,
+
u
v
,
μ
N
i
∞
,
-
u
v
>
μ
N
i
′
∞
,
-
u
v
for  some
u
,
v
∈
U
.Edge x
y is an N
i-P bipolar fuzzy bridge if only the first condition holds and an N
i-N bipolar fuzzy bridge if only the second condition holds.Example 35.
Consider the BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
) as shown in Figure 6 and let G
ˇ
b
′
=
(
M
,
N
1
′
,
N
2
′
) be bipolar fuzzy spanning subgraph structure of G
ˇ
b obtained by deleting N
1-edge (
a
2
a
5
). Then a
2
a
5 is a bipolar fuzzy N
1-bridge, since μ
N
1
∞
,
+
(
a
2
a
5
)
=
0.4
>
0.3
=
μ
N
1
′
∞
,
+
(
a
2
a
5
) and μ
N
1
∞
,
-
(
a
2
a
5
)
=
0.4
>
0.3
=
μ
N
1
′
∞
,
-
(
a
2
a
5
), and alsoabipolar fuzzy
N
1-N bridge, since μ
N
1
∞
,
-
(
a
3
a
5
)
=
0.4
>
0.3
=
μ
N
1
′
∞
,
-
(
a
3
a
5
) and μ
N
1
∞
,
-
(
a
4
a
5
)
=
0.4
>
0.3
=
μ
N
1
′
∞
,
-
(
a
4
a
5
)
.Definition 36.
A BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) of agraph structure
G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
) is an N
i-tree if (
supp
⁡
A
,
supp
⁡
N
1
,
supp
⁡
N
2
,
…
,
supp
⁡
(
N
n
)
) is an E
i-tree. In other words, G
ˇ
b is an N
i-tree if a subgraph of G
ˇ
b, induced by supp
⁡
(
N
i
), forms a tree.Definition 37.
A BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) of agraph structure
G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
) is abipolar fuzzy
N
i-tree if G
ˇ
b has abipolar fuzzy spanning subgraph structure
H
ˇ
b
=
(
A
,
C
1
,
C
2
,
…
,
C
n
) such that H
ˇ
b is a C
i-tree and μ
N
i
P
(
x
y
)
<
μ
C
i
∞
,
+
(
x
y
) and μ
N
i
N
x
y
<
μ
C
i
∞
,
-
(
x
y
)
∀
N
i-edges not in H
ˇ
b
.
In more concerned view,G
ˇ
b is abipolar fuzzy
N
i
-P tree if only the first condition holds and abipolar fuzzy
N
i-N tree if only the second condition holds.Example 38.
Consider BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
) as shown in Figure 7, which is an N
2-tree. It is not an N
1-tree but abipolar fuzzy
N
1-tree since it has a bipolar fuzzy spanning subgraph structure (
M
,
N
1
′
,
N
2
′
) as an N
1-tree, which is obtained by deleting N
1-edge a
2
a
5 from G
ˇ
b and(28)
μ
N
1
P
a
2
a
5
=
0.2
<
0.3
=
μ
N
1
′
∞
,
+
a
2
a
5
,
μ
N
1
N
a
2
a
5
=
0.1
<
0.3
=
μ
N
1
′
∞
,
-
a
2
a
5
.Figure 7
G
ˇ
b
=
(
M
,
N
1
,
N
2
).Definition 39.
A BFGSG
ˇ
s
1
=
(
M
1
,
N
11
,
N
12
,
…
,
N
1
n
) ofgraph structure
G
1
∗
=
(
U
1
,
E
11
,
E
12
,
…
,
E
1
n
) isisomorphic to a BFGS G
ˇ
s
2
=
(
M
2
,
N
21
,
N
22
,
…
,
N
2
n
) of G
2
∗
=
(
U
2
,
E
21
,
E
22
,
…
,
E
2
n
) if there exists a bijection f
:
U
1
→
U
2 and a permutation ϕ on the set {
1,2
,
…
,
n
} such that(29)
μ
M
1
P
u
1
=
μ
M
2
P
f
u
1
,
μ
M
1
N
u
1
=
μ
M
2
N
f
u
1
∀
u
1
∈
U
1and for ϕ
(
i
)
=
j
(30)
μ
N
1
i
P
u
1
u
2
=
μ
N
2
j
P
f
u
1
f
u
2
,
μ
N
1
i
N
u
1
u
2
=
μ
N
2
j
N
f
u
1
f
u
2
∀
u
1
u
2
∈
E
1
i
,
i
=
1,2
,
…
,
n
.Example 40.
LetG
ˇ
b
1
=
(
M
,
N
1
,
N
2
) and G
ˇ
b
2
=
(
M
′
,
N
1
′
,
N
2
′
) be two BFGSs of graph structures G
1
∗
=
(
U
,
E
1
,
E
2
) and G
2
∗
=
(
U
′
,
E
1
′
,
E
2
′
), respectively, as shown in Figure 8.
HereG
ˇ
b
1 is isomorphic (not identical) to G
ˇ
b
2 under the mapping f
:
U
→
U
′, defined by f
(
a
1
)
=
b
1, f
(
a
2
)
=
b
2, and f
(
a
3
)
=
b
3, and a permutation ϕ given by ϕ
(
1
)
=
2, ϕ
(
2
)
=
1, such that(31)
μ
M
P
a
i
=
μ
M
′
P
f
a
i
,
μ
M
N
a
i
=
μ
M
′
N
f
a
i
∀
a
i
∈
U
,
μ
N
k
P
a
i
a
j
=
μ
N
ϕ
k
P
f
a
i
f
a
j
,
μ
N
k
N
a
i
a
j
=
μ
N
ϕ
k
N
f
a
i
f
a
j
∀
a
i
a
j
∈
E
k
,
k
=
1,2
.Figure 8
Isomorphic bipolar fuzzy graph structures.Definition 41.
A BFGSG
ˇ
s
1
=
(
M
1
,
N
11
,
N
12
,
…
,
N
1
n
) of GS G
1
∗
=
(
U
,
E
11
,
E
12
,
…
,
E
1
n
) isidentical to a BFGS G
ˇ
s
2
=
(
M
2
,
N
21
,
N
22
,
…
,
N
2
n
) of GS G
2
∗
=
(
U
,
E
21
,
E
22
,
…
,
E
2
n
) if there exist a bijection f
:
U
→
U, such that(32)
μ
M
1
P
u
=
μ
M
2
P
f
u
,
μ
M
1
N
u
=
μ
M
2
N
f
u
∀
u
∈
U
,
μ
N
1
i
P
u
1
u
2
=
μ
N
2
i
P
f
u
1
f
u
2
,
μ
N
1
i
N
u
1
u
2
=
μ
N
2
i
N
f
u
1
f
u
2
∀
u
1
u
2
∈
E
1
i
,
i
=
1,2
,
…
,
n
.Example 42.
LetG
ˇ
b
1
=
(
M
,
N
1
,
N
2
) and G
ˇ
b
2
=
(
M
′
,
N
1
′
,
N
2
′
) be two BFGSs of graph structures G
1
∗
=
(
U
,
E
1
,
E
2
) and G
2
∗
=
(
U
′
,
E
1
′
,
E
2
′
), respectively, as shown in Figure 9.
HereG
ˇ
b
1 is identical with G
ˇ
b
2 under the mapping f
:
U
→
U
′, defined by f
(
a
1
)
=
b
6
,
f
(
a
2
)
=
b
2
,
f
(
a
3
)
=
b
4
,
f
(
a
4
)
=
b
5
,
f
(
a
5
)
=
b
1, and f
(
a
6
)
=
b
3, such that(33)
μ
M
P
a
i
=
μ
M
′
P
f
a
i
,
μ
M
N
a
i
=
μ
M
′
N
f
a
i
∀
a
i
∈
U
,
μ
N
k
P
a
i
a
j
=
μ
N
k
′
P
f
a
i
f
a
j
,
μ
N
k
N
a
i
a
j
=
μ
N
k
′
N
f
a
i
f
a
j
∀
a
i
a
j
∈
E
k
,
k
=
1,2
.Figure 9
Identical bipolar fuzzy graph structures.Definition 43.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) be a BFGS of a GS G
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
)
. Let ϕ be any permutation on the set {
E
1
,
E
2
,
…
,
E
n
} and the corresponding permutation on {
N
1
,
N
2
,
…
,
N
n
}; that is, ϕ
(
N
i
)
=
N
j if and only if ϕ
(
E
i
)
=
E
j
∀
i
.
Ifx
y
∈
N
r for some r and (34)
μ
N
i
ϕ
P
x
y
=
μ
M
P
x
∧
μ
M
P
y
-
⋁
j
≠
i
μ
ϕ
N
j
P
x
y
,
μ
N
i
ϕ
N
x
y
=
μ
M
N
x
∨
μ
M
N
y
-
⋀
j
≠
i
μ
ϕ
N
j
N
x
y
,
i
=
1,2
,
…
,
n
,then x
y
∈
B
m
ϕ, while m is chosen such that μ
N
m
ϕ
P
(
x
y
)
≥
μ
N
i
ϕ
P
(
x
y
) and μ
N
m
ϕ
N
(
x
y
)
≤
μ
N
i
ϕ
N
(
x
y
)
∀
i
.
And BFGS(
M
,
N
1
ϕ
,
N
2
ϕ
,
…
,
N
n
ϕ
), denoted by G
ˇ
b
ϕ
c, is called the ϕ-complement of BFGS G
ˇ
b
.Example 44.
LetM
=
{
(
a
1
,
0.3
,
-
0.7
), (
a
2
,
0.5
,
-
0.4
), (
a
3
,
0.7
,
-
0.3
)
}, N
1
=
{
(
a
1
a
3
,
0.3
,
-
0.3
)
,
(
a
2
a
3
,
0.5
,
-
0.3
)
}, and N
2
=
{
(
a
1
a
2
,
0.3
,
-
0.4
)
} be bipolar fuzzy subsets of U
,
E
1, and E
2, respectively, so that G
ˇ
b
=
(
M
,
N
1
,
N
2
) is a BFGS of graph structure G
∗
=
(
U
,
E
1
,
E
2
)
. Let ϕ be a permutation on the set {
N
1
,
N
2
} such that ϕ
(
N
1
)
=
N
2 and ϕ
(
N
2
)
=
N
1
.
Now fora
2
a
3
∈
N
1,(35)
μ
N
1
ϕ
P
a
2
a
3
=
μ
M
P
a
2
∧
μ
M
P
a
3
-
⋁
j
≠
1
μ
ϕ
N
j
P
a
2
a
3
=
0.5
∧
0.7
-
μ
ϕ
N
2
P
a
2
a
3
=
0.5
-
μ
N
1
P
a
2
a
3
=
0.5
-
0.5
=
0
,
μ
N
1
ϕ
N
a
2
a
3
=
μ
M
N
a
2
∨
μ
M
N
a
3
-
⋀
j
≠
1
μ
ϕ
N
j
N
a
2
a
3
=
-
0.4
∨
-
0.3
-
μ
ϕ
N
2
N
a
2
a
3
=
-
0.3
-
μ
N
1
N
a
2
a
3
=
-
0.3
+
0.3
=
0
,
μ
N
2
ϕ
P
a
2
a
3
=
μ
M
P
a
2
∧
μ
M
P
a
3
-
⋁
j
≠
2
μ
ϕ
N
j
P
a
2
a
3
=
0.5
∧
0.7
-
μ
ϕ
N
1
P
a
2
a
3
=
0.5
-
μ
N
2
P
a
2
a
3
=
0.5
-
0
=
0.5
,
μ
N
2
ϕ
N
a
2
a
3
=
μ
M
N
a
2
∨
μ
M
N
a
3
-
⋀
j
≠
2
μ
ϕ
N
j
N
a
2
a
3
=
-
0.4
∨
-
0.3
-
μ
ϕ
N
1
N
a
2
a
3
=
-
0.3
-
μ
N
2
N
a
2
a
3
=
-
0.3
-
0
=
-
0.3
.Clearly, μ
ϕ
N
2
P
(
a
2
a
3
)
=
0.5
>
0
=
μ
ϕ
N
1
P
(
a
2
a
3
) and μ
ϕ
N
2
N
(
a
2
a
3
)
=
-
0.3
<
0
=
μ
ϕ
N
1
N
(
a
2
a
3
)
. So a
2
a
3
∈
N
2
ϕ.
Similarly fora
1
a
3
∈
N
1
,
μ
N
1
ϕ
P
(
a
1
a
3
)
=
0
,
μ
N
1
ϕ
N
(
a
1
a
3
)
=
0
,
μ
N
2
ϕ
P
(
a
1
a
3
)
=
0.3, and μ
N
2
ϕ
N
a
1
a
3
=
-
0.3
.
⇒
μ
N
2
ϕ
P
(
a
1
a
3
)
=
0.3
>
0
=
μ
N
1
ϕ
P
(
a
1
a
3
) and μ
N
2
ϕ
N
(
a
1
a
3
)
=
-
0.3
<
0
=
μ
N
1
ϕ
N
(
a
1
a
3
). So a
1
a
3
∈
N
2
ϕ.
And fora
1
a
2
∈
N
2
,
μ
N
1
ϕ
P
(
a
1
a
2
)
=
0.3
,
μ
N
1
ϕ
N
(
a
1
a
2
)
=
-
0.4
,
μ
N
2
ϕ
P
(
a
1
a
2
)
=
0, and μ
N
2
ϕ
N
(
a
1
a
2
)
=
0
.
⇒
μ
N
1
ϕ
P
(
a
1
a
2
)
=
0.3
>
0
=
μ
N
2
ϕ
P
(
a
1
a
2
) and μ
N
1
ϕ
N
(
a
1
a
2
)
=
-
0.4
<
0
=
μ
N
2
ϕ
N
(
a
1
a
2
). So a
1
a
2
∈
N
1
ϕ.
This implies that(36)
N
1
ϕ
=
a
1
a
2
,
0.3
,
-
0.4
,
N
2
ϕ
=
a
2
a
3
,
0.5
,
-
0.3
,
a
1
a
3
,
0.3
,
-
0.3and G
ˇ
b
ϕ
c
=
(
M
,
N
1
ϕ
,
N
2
ϕ
) is the ϕ-complement of G
ˇ
b
.Theorem 45.
Aϕ-complement of a bipolar fuzzy graph structure is always a strong BFGS. Moreover, if ϕ
(
i
)
=
r for r
,
i
∈
{
1,2
,
…
,
n
}, then all N
r-edges in BFGS G
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) become B
i
ϕ-edges in G
ˇ
b
ϕ
c
=
(
A
,
B
1
ϕ
,
B
2
ϕ
,
…
,
B
n
ϕ
)
.Proof.
From the definition ofϕ-complement G
ˇ
b
ϕ
c,(37)
μ
N
i
ϕ
P
x
y
=
μ
M
P
x
∧
μ
M
P
y
-
⋁
j
≠
i
μ
ϕ
N
j
P
x
y
,
(38)
μ
N
i
ϕ
N
x
y
=
μ
M
N
x
∨
μ
M
N
y
-
⋀
j
≠
i
μ
ϕ
N
j
N
x
y
,for i
=
1,2
,
…
,
n
.
Let us consider expression (37) first.
Sinceμ
M
N
(
x
)
∨
μ
M
N
(
y
)
≤
0 and ⋀
j
≠
i
μ
ϕ
N
j
N
(
x
y
)
≤
0, we can write (39)
μ
N
i
ϕ
N
x
y
=
-
μ
M
N
x
∨
μ
M
N
y
+
⋀
j
≠
i
μ
ϕ
N
j
N
x
y
.Also from the definition of a BFGS μ
N
j
N
x
y
≥
μ
M
N
x
∨
μ
M
N
y
∀
N
j
(40)
⟹
⋀
j
≠
i
μ
ϕ
N
j
N
x
y
≥
μ
M
N
x
∨
μ
M
N
y
⟹
⋀
j
≠
i
μ
ϕ
N
j
N
x
y
≤
μ
M
N
x
∨
μ
M
N
y
⟹
-
μ
M
N
x
∨
μ
M
N
y
+
⋀
j
≠
i
μ
ϕ
N
j
N
x
y
≤
0
.Therefore, μ
N
i
ϕ
N
(
x
y
)
≤
0
∀
i.
Now a requirement is minimum value ofμ
N
i
ϕ
N
(
x
y
)
. Since μ
N
i
ϕ
N
(
x
y
)
≤
0, that is why it is minimum when its positive part ⋀
j
≠
i
μ
ϕ
N
j
N
x
y is zero. And ⋀
j
≠
i
μ
ϕ
N
j
N
x
y
=
0 when ϕ
N
i
=
N
r and x
y is an N
r-edge. So(41)
μ
N
i
ϕ
N
x
y
=
μ
M
N
x
∨
μ
M
N
y
,
for
x
y
∈
N
r
,
ϕ
N
i
=
N
r
.Similarly for expression (38), a requirement is maximum value of μ
N
i
ϕ
P
x
y
. Since μ
M
P
x
∧
μ
M
P
y
≥
0, ⋁
j
≠
i
μ
ϕ
N
j
P
x
y
≥
0 and μ
N
j
P
x
y
≤
μ
M
P
x
∧
μ
M
P
y
∀
N
j
(42)
⟹
⋁
j
≠
i
μ
ϕ
N
j
P
x
y
≤
μ
M
P
x
∧
μ
M
P
y
⟹
μ
M
P
x
∧
μ
M
P
y
-
⋁
j
≠
i
μ
ϕ
N
j
P
x
y
≥
0
.Therefore, μ
N
i
ϕ
P
(
x
y
)
≥
0
∀
i.
Nowμ
N
i
ϕ
P
(
x
y
) will be maximum when its negative part [
-
⋁
j
≠
i
μ
ϕ
N
j
P
(
x
y
)
] becomes zero. Clearly, [
-
⋁
j
≠
i
μ
ϕ
N
j
P
(
x
y
)
]
=
0 when ϕ
N
i
=
N
r and x
y is an N
r-edge. So(43)
μ
N
i
ϕ
P
x
y
=
μ
M
P
x
∧
μ
M
P
y
,
for
x
y
∈
N
r
,
ϕ
N
i
=
N
r
.From (41) and (43), the conclusion is obvious.Definition 46.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) be a BFGS and let ϕ be any permutation on the set {
1,2
,
…
,
n
}
. Then(i)
G
ˇ
b isself-complement if it is isomorphic to G
ˇ
b
ϕ
c, the ϕ-complement of G
ˇ
b;
(ii)
G
ˇ
b isstrong self-complement if it isidentical to G
ˇ
b
ϕ
c.Definition 47.
LetG
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) be a BFGS. Then (i)
G
ˇ
b istotally self-complement if it isisomorphic to G
ˇ
b
ϕ
c, the ϕ
-complement of G
ˇ
b, for all permutations ϕ on the set {
1,2
,
…
,
n
};
(ii)
G
ˇ
b istotally strong self-complement if it isidentical to G
ˇ
b
ϕ
c, the ϕ-complement of G
ˇ
b, for all permutations ϕ on the set {
1,2
,
…
,
n
}
.Example 48.
All strong BFGSs are the only examples of self-complement or totally self-complement BFGSs.Example 49.
A BFGSG
ˇ
b
=
(
M
,
N
1
,
N
2
,
N
3
) of graph structure G
∗
=
(
U
,
E
1
,
E
2
,
E
3
) as shown in Figure 10 is totally strong self-complement.Figure 10
Totally strong self-complement BFGS.Theorem 50.
A BFGSG
ˇ
b is strong if and only if it is totally self-complement.Proof.
LetG
ˇ
b be a strong BFGS and ϕ any permutation on the set {
1,2
,
…
,
n
}.
By Theorem45, G
ˇ
b
ϕ
c is strong and if ϕ
-
1
(
i
)
=
j, then all N
i-edges in G
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) become N
j
ϕ-edges in G
ˇ
b
ϕ
c
=
M
,
N
1
ϕ
,
N
2
ϕ
,
…
,
N
n
ϕ
(44)
⟹
μ
N
i
P
a
1
a
2
=
μ
M
P
a
1
∧
μ
M
P
a
2
=
μ
N
j
ϕ
P
a
1
a
2
,
μ
N
i
N
a
1
a
2
=
μ
M
N
a
1
∨
μ
M
N
a
2
=
μ
N
j
ϕ
N
a
1
a
2
.Hence G
ˇ
b is isomorphic to G
ˇ
b
ϕ
c under the identity mapping f
:
U
→
U, such that μ
M
P
(
a
)
=
μ
M
P
(
f
(
a
)
)
,
μ
M
N
(
a
)
=
μ
M
N
(
f
(
a
)
)
∀
a
∈
U and(45)
μ
N
i
P
a
1
a
2
=
μ
N
j
ϕ
P
a
1
a
2
=
μ
N
j
ϕ
P
f
a
1
f
a
2
,
μ
N
i
N
a
1
a
2
=
μ
N
j
ϕ
N
a
1
a
2
=
μ
N
j
ϕ
N
f
a
1
f
a
2
,
∀
a
1
a
2
∈
E
i
,for ϕ
-
1
i
=
j
,
i
,
j
=
1,2
,
…
,
n
. This holds for any permutation on the set {
1,2
,
…
,
n
}
.
HenceG
ˇ
b is totally self-complement.
Conversely, letG
ˇ
b and G
ˇ
b
ϕ
c be isomorphic for any permutation ϕ on the set {
1,2
,
…
,
n
}
. Then from the definition of ϕ-complement and isomorphism of BFGSs, we have(46)
μ
N
i
P
a
1
a
2
=
μ
N
j
ϕ
P
f
a
1
f
a
2
=
μ
M
P
f
a
1
∧
μ
M
P
f
a
2
=
μ
M
P
a
1
∧
μ
M
P
a
2
,
μ
N
i
N
a
1
a
2
=
μ
N
j
ϕ
N
f
a
1
f
a
2
=
μ
M
N
f
a
1
∨
μ
M
N
f
a
2
=
μ
M
N
a
1
∨
μ
M
N
a
2
∀
a
1
a
2
∈
E
i
,
i
=
1,2
,
…
,
n.
Hence,G
ˇ
b is a strong BFGS.Remark 51.
Everyself-complement BFGS is necessarilytotally self-complement.Theorem 52.
If graph structureG
∗
=
(
U
,
E
1
,
E
2
,
…
,
E
n
) is totally strongself-complement and M
=
(
μ
M
P
,
μ
M
N
) is a bipolar fuzzy set of U with constant valued functions μ
M
P and μ
M
N, then a strong BFGS G
ˇ
b
=
(
M
,
N
1
,
N
2
,
…
,
N
n
) of G
∗ istotally strong self-complement.Proof.
Lets
∈
[
0,1
] and t
∈
[
-
1,0
] be two constants, such that (47)
μ
M
P
u
=
s
,
μ
M
N
u
=
t
∀
u
∈
U
.Since G
∗ is totally strong self-complement, so for every permutation ϕ
-
1 on the set {
1,2
,
…
,
n
}, there exists a bijection f
:
U
→
U, such that for every E
i-edge a
1
a
2, “f
(
a
1
)
f
(
a
2
)” [an E
j-edge in G
∗] is an E
i-edge in (
G
∗
)
ϕ
-
1
c and, consequently, for every N
i-edge a
1
a
2, “f
(
a
1
)
f
(
a
2
)” [a N
j-edge in G
ˇ
b] is a B
i
ϕ-edge in G
ˇ
b
ϕ
c
. Moreover G
ˇ
b is strong, so we have(48)
μ
M
P
a
=
s
=
μ
M
P
f
a
,
μ
M
N
a
=
t
=
μ
M
N
f
a
∀
u
∈
U
,
μ
N
i
P
a
1
a
2
=
μ
M
P
a
1
∧
μ
M
P
a
2
=
μ
M
P
f
a
1
∧
μ
M
P
f
a
2
=
μ
N
j
ϕ
P
f
a
1
f
a
2
,
μ
N
i
N
a
1
a
2
=
μ
M
N
a
1
∨
μ
M
N
a
2
=
μ
M
N
f
a
1
∨
μ
M
N
f
a
2
=
μ
N
j
ϕ
N
f
a
1
f
a
2
∀
a
1
a
2
∈
E
i
,
i
=
1,2
,
…
,
n.
This shows thatG
ˇ
b is strong self-complement. This holds for any permutation ϕ and ϕ
-
1 on the set {
1,2
,
…
,
n
}; thus G
ˇ
b is totally strong self-complement. This completes the proof.Remark 53.
The converse of Theorem52 is not necessary, since atotally strong self-complement BFGS
G
ˇ
b
=
(
M
,
N
1
,
N
2
,
N
3
), as shown in Figure 10, isstrong and has atotally strong self-complement underlying graph structure, but μ
M
P and μ
M
N 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.



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*Source: 5859080-2016-02-10.xml*