Seetha Madam Maths

Science and Information Conference 2013 October 7-9, 2013 | London, UK

557 | P a g e www.conference.thesai.org

Concept of Fuzzy Planar Graphs

Anita Pal Department of Mathematics

National Institute of Technology Durgapur

West Bengal, India.

Sovan Samanta and Madhumangal Pal

Department of Applied Mathematics with Oceanology and

Computer Programming, Vidyasagar University,

Midnapore - 721 102, India.

Abstract—Fuzzy graph is now a very important research area

due to its wide application. Fuzzy multigraph and fuzzy planar

graphs are two subclasses of fuzzy graph theory. In this paper, we

define both of these graphs and studied a lot of properties. A very

close association of fuzzy planar graph is fuzzy dual graph. This is

also defined and studied several properties. The relation between

fuzzy planar graph and fuzzy dual graph is also established.

Keywords—Fuzzy graphs; fuzzy multigraphs; fuzzy planar

graphs; fuzzy dual graphs

I. INTRODUCTION

Graph theory has vast applications in computer science like data mining, image segmentation, clustering, image capturing, networking, etc. For example, a data structure can be designed in the form of a tree which in turn utilizes vertices and edges. Similarly, modeling of network topologies can be done using graph concepts. In the same way the most important concept of graph colouring is utilized in resource allocation, scheduling, etc. Also, paths, walks and circuits in graph theory are used in other applications, viz. traveling salesman problem, database design concepts, resource networking, etc. This leads to the development of new algorithms and new theories that can be used in various applications.

There are also many practical applications with a graph structure in which crossing edges are a nuisance, including design problems for circuits, subways, utility lines. Two crossing connections normally mean that the edges must be run at different heights. This is not a big issue for electrical wires, but it creates extra expenses for some types of lines, e.g. burying one subway tunnel under another (and therefore deeper than it would ordinarily need). Circuits, in particular, are easier to manufacture if their connections can be constructed in fewer layers. These applications are designed by the concept of planar graphs. Numerous computational challenges like image segmentation or shape matching can be solved by means of planar graph cuts.

The famous utility problem is that of connecting gas, water, electric line to three houses without crossing each other. This problem and many other problems cannot be solved by planar graph model. These problems motivate us to define fuzzy planar graphs.

Fuzzy graph theory has many applications in modern science. It is used in evaluation of human cardiac function, fuzzy neural networks, etc. A fuzzy graph is also used to solve traffic light problem, time table scheduling, etc. A lot of work has been done on fuzzy graphs [2, 3, 7, 8, 9, 11, 13].

To the best of our knowledge no works have been done on fuzzy multigraphs, fuzzy planar graphs and fuzzy dual graphs. In this paper, we introduced these graphs for first time, illustrated by examples and established some results.

A. Review of literature

After development of fuzzy graph by Rosenfeld [24], the fuzzy graph theory is increased with a large number of branches. Samanta and Pal introduced fuzzy tolerance graphs [21], fuzzy threshold graphs [22] and fuzzy competition graphs [23]. Nagoorgani and Radha [14] defined regular fuzzy graphs. On the other hand, planar graph theory is an important research area. In 1930, Kuratowski [10] invented some important results on planar graphs. Whitney’s planarity criterion gives a characterization based on the existence of an algebraic dual. Fraysseix–Rosenstiehl’s planarity criterion [6] gives a characterization based on the existence of a bipartition of the co-tree edges of a depth-first search tree. In this paper, we introduced the planarity of graphs in fuzzy sense.

II. PRELIMINARIES

A. Planar graphs

A finite graph ),(= EVG such that the set of vertices V

and set of edges E are finite. An infinite graph is one with an infinite set of vertices or edges or both. Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated.

A multigraph [1] is a graph that may contain multiple edges between any two vertices, but it does not contain any self loops.

A drawing of a geometric representation of a graph on any surface such that no edges intersect is called embedding [1]. A

graph G is planar if it can be drawn in the plane with its edges only intersecting at vertices of G. So the graph is non-planar if it cannot be drawn without crossing of edges. A planar graph with cycles divides the plane into a set of regions,

also called faces. The length of a face in a plane graph G is

the total length of the closed walk(s) in G bounding the face. The portion of the plane lying outside a graph embedded in a plane is infinite region. A region is characterized by the set of edges (or the set of vertices) forming its boundary. A region is not defined for non-planar graphs or even in planar graph not embedded in a plane. The portion of the plane lying outside a graph embedded in a plane is infinite in its extent, such a region is called infinite, unbounded, outer, or exterior for that particular plane representation.



In the graph theory, the dual graph of a given planar graph G is a graph which has a vertex corresponding to each plane region of G, and an edge joining two neighboring regions for each edge in G, for a certain embedding of G. The term “dual" is used because this property is symmetric, meaning that if H is a dual of G, then G is a dual of H (if G is connected).

If n, e and f denote as usual the numbers of vertices, edges

and regions of a connected graph G and if n’, e’, f’ are the

corresponding numbers in dual graph G’, n’=f, e’=e and

f’= n.

B. Fuzzy graphs

A fuzzy set A on a set X is characterized by a mapping m : X→[0,1], which is called the member -ship function. A fuzzy set is denoted by A=(X, m).

Definition 1 [31] A fuzzy graph ),,(= V is a

non-empty set V together with a pair of functions

[0,1]: V and [0,1]: VV such that for all

Vyx , , )()(),( yxyx , where )(x and

),( yx represent the membership value of the vertex x

and that of the edge ),( yx in and }.,min{ yxyx The function is a symmetric fuzzy relation on .

Since is well defined, a fuzzy graph has no multiple

edges. A loop at a vertex x in a fuzzy graph is represented

by 0),( xx . The fuzzy set ),( V is called fuzzy

vertex set of and the elements of the fuzzy set are called

fuzzy vertices. ),( VV is called fuzzy edge set of

and the elements of the fuzzy set are called fuzzy edges. An

edge is non-trivial if 0),( yx .

For the fuzzy graph ),,(= V , two vertices x and y

in V are called adjacent if 0.5 min{σ(x), σ(y)} ≤ µ(x, y). The

edge (x, y) of is called strong [7] if x and y are adjacent

and it called weak otherwise.

A fuzzy graph ),,(= V is complete if

)()(=),( vuvu for all Vvu ,

where ),( vu

denotes the edge between u and v .

The complement of a fuzzy graph ),,( V is defined

as a fuzzy graph ),,( ccV where σc

and c are given

by σc(v) = σ(v) for all Vv and

.0=),(),()(

0>),(0,

=),(

vuifvu

vuif

vuc

A fuzzy graph ),,(= V is said to be bipartite if the

vertex set V can be partitioned into two nonempty sets 1V

and 2V such that 0=),( 21 vv if

121, Vvv or

., 221 Vvv Further if )()(=),( 2121 vvvv for all

11 Vv and 22 Vv then is called a complete bipartite

graph.

An edge e = (x,y) of a fuzzy graph is called an effective edge [20] if .)()(=),( yxyx

Two vertices are

said to be effective adjacent [20] if they are the end vertices

of the same effective edge. Let ),,(= V be a fuzzy

graph. Then the effective incident degree [20] of a fuzzy graph is defined as number of effective incident edges on a

vertex v . A pendant vertex [20] in a fuzzy graph is defined as a vertex of an effective incident degree one. Fuzzy pendent edge [23] in a fuzzy graph is a fuzzy edge whose one end vertex is a fuzzy pendent vertex. So the membership value of the pendent edge is the minimum membership values of the end vertices.

The underlying crisp graph of the fuzzy graph

),,(= V is denoted as

),,(= *** V where

0})(|{=* uVu and

.0}),(|),{(=* vuVVvu

C. Fuzzy multisets

A (crisp) multiset over a non-empty set V is simply a

mapping NVd : where N is the set of natural numbers. Yager [25] first discussed fuzzy multisets, although

he uses the term of fuzzy bag, an element of nonempty set V may occur more than once with possibly the same or different membership values. A natural generalization of this interpretation of multisets leads to the notion of fuzzy multiset, or fuzzy bag, over a non-empty set V as a

mapping NVC [0,1]:~

. The membership values of

Vv are denoted as pjv j ,1,2,=,

where

0}.:{= jvjmaxp

So the fuzzy multiset can be

denoted by

}|,1,2,=),,{(= VvpjvvM j

.

For Vv , the membership sequence is defined to be the

decreasingly-ordered sequence of the membership values of the

elements [0,1]}}{{x domain of C~

. It is denoted by

),,,( 21 pvvv

where pvvv

21

.

Now we define fuzzy multigraph. In this graph fuzzy multiedges are allowed.

III. FUZZY MULTIGRAPHS

Definition 2. Let V be a non-empty set and

[0,1]: V be a mapping. Also let

}),(|,1,2,=),),(),,{((= VVyxpjyxyxE j

be a fuzzy multiset of VV such that

)()(),( yxyx j

for all pj ,1,2,= where



.0}),(|{= jyxjmaxp

Then

),,(= EV is denoted as fuzzy multigraph where )(x

and jyx

),( represent the membership value of the vertex

x and the membership value of the edge ),( yx in

respectively.

In fuzzy multigraph ),,(= EV , E is said to be

fuzzy multi-edge set. An example of fuzzy multigraph is given below.

Example 1. Let },,,{= zyxwV and [0,1]: V

such that

0.8=)(0.65,=)(0.7,=)(0.6,=)( zyxw . Let

.),0.8)},((),0.6),,((

),0.6),,((),0.55),,((),0.5),,{((=

zxyx

xwxwxwEHere

),,( EV is a fuzzy multigraph.

Degree of a vertex in fuzzy graph is the sum of membership values of edges adacent to the vertex. It is an important characteristic of a fuzzy multigraph also.

Definition 3. Let ),,(= EV be a fuzzy multigraph

where

}),(|,1,2,=),),(),,{((= VVyxpjyxyxE j

and

0}),(|{= jyxjmaxp

. The degree of a vertex Vy

is denoted by )(yd and is defined by j

p

j

yxyd

),(=)(1=

for all Vx .

Example 2. In Example 1, degree of the vertices

zyxw ,,, are 1.65=0.60.550.5=)( wd ,

.95.2=0.70.60.60.550.5=)( xd 0.6=)(yd , 0.8=)(zd .

The concept of strong edges in the fuzzy graphs is now an important term in fuzzy graph theory. Strong edges in fuzzy multigraph is defined below.

Definition 4. Let ),,(= EV be a fuzzy multigraph,

where

}),(|,1,2,=),),(),,{((= VVyxpjyxyxE j

and

0}),(|{= jyxjmaxp

. A multiedge (x, y) of is

strong if

jyxyxmin

),()}(),({2

1 for all

pj ,1,2,= .

An edge )),(),,(( yxyx in fuzzy multigraph is strong

edge if

.)}(),({2

1),( yxminyx

Example 3. In Example 1,

0.6=),(0.65,=)(0.7,=)( yxyx

and hence min{0.65, 0.7)/2 <0.6. So ),( yx is a strong edge.

Complete graph is a special kind of graph in graph theory where all vertices are connected with other vertices by edges. Fuzzy complete multigraph is defined below.

Definition 5. Let ),,(= EV be a fuzzy multigraph

where

}),(|,1,2,=),),(),,{((= VVyxpjyxyxE j

and

0}),(|{= jyxjmaxp

. is called fuzzy complete

multigraph if

)()(=),( yxyx j

for all pj ,1,2,=

and for all Vyx , .

Example 4. Let },,{= 321 vvvV ,

0.4=)(0.7,=)(0.5,=)( 321 vvv and

),0.4)},((),0.5),,((),0.5),,{((= 322121 vvvvvvE . So

0.4=),(,),(=0.5=),( 132221121 vvvvvv

.0.4=),( 131 vv

Here ),,( EV is an example of fuzzy complete

multigraph.

IV. FUZZY PLANAR GRAPHS

A crisp graph is called non-planar graph if there is at least one crossing for all geometrical representations of the graph.

Let a crisp graph G has a crossing for a certain geometrical

representation between two edges ),( ba and ),( dc . In

fuzzy concept, we say that this two edges have membership

values 1. If we remove the edge ),( dc , the graph becomes

planar. In fuzzy sense, we say that the edges ),( ba and

),( dc have membership values 1 and 0 respectively.

Let ),,(= V be a fuzzy graph and for a certain

geometric representation, the graph has only one crossing between two fuzzy edges )),(),,(( xwxw and

)),(),,(( zyzy . If 1=),( xw and 0=),( zy then

we say that the fuzzy graph has no crossing. Similarly, if

),( zy has value near to 1 and ),( xw has value near to

0 , the crossing will not be important for the representation of

the graph. If ),( xw has value near to 1 and ),( xw

has value near to 1, then the crossing will be important for the representation of the graph.

Before going to the main definition, a co-related term is discussed below.

A. Intersecting value in fuzzy multigraph

Strong property of the fuzzy edge ),( ba can be measured

by the value



)()(

),(=),(

ba

baI

k

ba

. If ,0.5),( baI the

fuzzy edge is strong otherwise weak.

In fuzzy multigraph, when two edges intersect at a point, a value is assigned to that point in the following way. Let in a fuzzy multigraph ),,(= EV , E contains two edges

)),(),,(( kbaba

and )),(),,(( ldcdc

which are

intersected at a point P, where k, l are fixed integers.

We define the intersecting value at the point P by IP

2=

),(),( dcba II . If the number of points of intersection in

fuzzy multigraph increases, planarity decreases. So for fuzzy multigraph, Ip is inversely proportional to the planarity.

Definition 6. Let be a fuzzy multigraph and for a certain

geometrical representation the fuzzy multigraph has

intersecting points zPPP ,,, 21 . is said to be fuzzy

planar graph with fuzzy planarity value f , where

.}{1

1=

21 zPPP

fIII

It is obvious that f is bounded and it is 1<0 f .

If there is no point of intersection for a certain geometrical representation of a fuzzy planar graph then its fuzzy planarity value is assigned to 1. The underlying crisp graph of this fuzzy graph is the crisp planar graph.

Example 5. Let us consider the fuzzy planar graph of

Figure 1. Here },,,,{= edcbaV be the set of vertices with

0.9,=)(0.6,=)(0.7,=)(0.8,=)( dcba

,0.9=)(e

0.55=),(0.4,=),(0.3,=),(

0.45,=),(0.5,=),(0.6,=),(

112

111

dbcbda

dacaba

.0.7=),(0.8,=),(

0.6,=),(0.5,=),(0.8,=),(

12

111

bede

dceaea

The fuzzy multigraph, ),,( EV has two points of

intersections P1 and P2. P1 is between the edges ((a,d), 0.45) and ((b,c), 0.4) and P2 is between ((a,d), 0.3) and ((b,c), 0.4). For the edge ((a,d), 0.45), I(a,d) = 0.45/0.8 = 0.5625 and for the edge ((b,c), 0.4), I(b,c) = 0.4/0.6 = 0.6667. For the first point of intersection P1, the intersection value IP1 is 0.6146 and that for the second point of intersection P2 that is 0.5208. So the fuzzy planarity value for the fuzzy multigraph shown in Figure 1 is 0.4683.

Fuzzy planarity value for the fuzzy complete multigraph is calculated from the following theorem.

Fig. 1. Example of fuzzy planar graph with fuzzy planarity value 0.4683

Theorem 1. Let ψ be a fuzzy complete multigraph. The fuzzy

planarity value, f of ψ is given by

inedgestheofonsintersectiofnumber

f

1

1

=

Proof. Let ),,(= EV be a fuzzy complete multigraph

where

}),(|,1,2,=),),(),,{((= VVyxpjyxyxE j

and

0}),(|{= jyxjmaxp

.

For fuzzy complete multigraph

)()(=),( yxyx j

for each j =1, 2, …, p.

Let kPPP ,,, 21 , k an integer, be the points of

intersections of the edges of . For any edge ),( ba in

fuzzy complete multigraph, 1=)()(

),(=),(

ba

baI

j

ba

. So

for 1P , the point of intersection between the edges ),( ba

and ),( dc is 1=2

11=

1

PI

. Hence 1=i

PI for

.,1,2,= ki

Now,

inedgesofonsintersectiofnumber

f

kPPP

1

1=

1)1(11

1=

1

1=

21

III

It follows from definition that any fuzzy multigraph is a fuzzy planar graph with some fuzzy planarity value. Now we introduce a subclass of fuzzy planar graph, namely strong fuzzy planar graph.

Definition 7. A fuzzy planar graph ψ is called strong fuzzy planar graph if fuzzy planarity value of the graph is greater than 0.5.



In Example 5, the fuzzy planar graph is not strong fuzzy planar graph as its fuzzy planarity value is less than 0.5.

Theorem 2. Let ψ be a strong fuzzy planar graph, then the

maximum number of point of intersections between strong

edges in ψ is one.

Theorem 3. Let ψ

be a strong fuzzy planar graph. The

number of point of intersections between weak edges in ψ may

be more than two.

In crisp sense, K5 and K3,3 cannot be drawn without

crossings. So any graph containing K5 or K3,3 is non-planar.

Theorem 4. A fuzzy complete graph of five vertices is not

strong fuzzy planar graph.

Proof. Let ),,(= EV be a fuzzy complete graph of five

vertices where },,,,{= edcbaV and

}),(|)),(),,{((= VVyxyxyxE . For all Vyx , ,

)()(=),( yxyx .

From the Theorem 1, the fuzzy planarity value of fuzzy complete graph is

inedgesofonsintersectiofnumberf

1

1=

We know that the geometric representation of the underlying crisp graph of fuzzy complete graph of five vertices is non-planar [1] and one point of intersection can not be

avoided for any representation. So .0.5=11

1=

f Hence

is not strong fuzzy planar graph. ∎

A fuzzy graph with five vertices and each pair of vertices connected by a fuzzy edge may or may not be strong fuzzy planar graph. In Figure 2, there is one point of intersection between two fuzzy edges ((v2,v4), 0.15), ((v1,v3), 0.45). This is an example of strong fuzzy planar graph with fuzzy planarity value 0.63492, which has five vertices and each pair of vertices are connected by a fuzzy edge.

Theorem 5. Complete bipartite fuzzy graph K3,3 is not strong fuzzy planar graph.

A fuzzy bipartite graph of six vertices can be partitioned in two subsets containing three vertices each and all pairs of vertices between two partitioned sets are connected by fuzzy edges may or may not be strong fuzzy planar graph. From Theorem 5, we know that a complete bipartite fuzzy graph of six vertices is not strong fuzzy planar graph. Hence the conclusion is true.

Fig. 2. Example of a strong fuzzy planar graph

Face of a planar graph is an important parameter. We now introduce the fuzzy face of a fuzzy planar graph.

Fuzzy face in a fuzzy graph is a region bounded by fuzzy edges. Every fuzzy face is characterized by fuzzy edges in its boundary. If all the edges in the boundary of a fuzzy face have membership value 1, it is called crisp face. If at least one of the edges of the boundary is fuzzy (membership value less than 1), then the face is called fuzzy face. Thus the existence of a fuzzy face depends on the existence of the fuzzy edge in the boundary.

Definition 8. Let ),,(= EV be a fuzzy planar graph

and

.}),(|,1,2,=),),(),,{((= VVyxpjyxyxE j

A fuzzy face of ψ is a region, bounded by the set of fuzzy

edges EE' , of a geometric representation of ψ. The membership value of the fuzzy face is given by

}),(|,1,2,=,)()(

),({ '

j

Eyxpjyx

yxmin

.

A fuzzy face is called strong fuzzy face if its membership value is greater than 0.5, and weak face otherwise. Every fuzzy planar graph has an infinite region which is called outer fuzzy face. Other faces are called inner fuzzy faces.

Example 6. In Figure 3, F1, F2 and F3 are three fuzzy faces.

F1 is bounded by the edges ((v1,v2), 0.5), ((v2,v3), 0.6),

((v1, v3), 0.55) with membership value 0.833. Similarly, F3 is a

fuzzy face bounded by the edges ((v1,v2), 0.5), ((v2,v4), 0.6), ((v1, v4), 0.2) with membership value 0.33. F2 is the outer fuzzy face with membership value 0.33. So here F1 is strong fuzzy face and F2, F3 are weak fuzzy faces.

Every strong fuzzy face has membership value greater than 0.5. So every edge of a strong fuzzy face is a strong fuzzy edge.



Fig. 3. Example of faces in fuzzy planar graph.

We denote fuzzy strong planar graph with fuzzy planarity

value more than or equal to 0.67 as 0.67-fuzzy planar graph. It is clear that 0.67-fuzzy planar graph is important as it does not contain any point of intersection between two strong edges. In crisp graph, face is related to only planar graphs. Here, strong fuzzy faces are related to 0.67-fuzzy planar graphs. All fuzzy planar graph without point of intersection of fuzzy edges is a fuzzy planar graph with fuzzy planarity value 1. So it is a 0.67-fuzzy planar graph.

V. FUZZY DUAL GRAPH

We now introduce dual of 0.67-fuzzy planar graph. In fuzzy dual graph, vertices are corresponding to the strong fuzzy faces of the 0.67-fuzzy planar graphs and each fuzzy edge between two vertices is corresponding to each edge in the boundary between two faces of 0.67-fuzzy planar graph. The formal definition is given below.

Definition 9. Let ),,(= EV be a 0.67-fuzzy planar

graph and

}),(|,1,2,=),),(),,{((= VVyxpjyxyxE j

. Also let

},,,{ 21 kFFF be the strong fuzzy faces of ψ. The fuzzy dual

graph of ψ is a fuzzy planar graph ),,(= '''' EV where

V’={xi, i=1, 2, …, k}.

The membership values of vertices are given by the

mapping [0,1]: '' V such that

),(|,1,2,=,),{(=)( vupjvumaxx ji

'

is an edge

of the boundary of the strong fuzzy face Fi }.

The membership values of the fuzzy edges of the fuzzy

dual graph are given by the membership values l

jlji vuxx

),(=),( where lvu ),( is an edge in the

boundary between two strong fuzzy faces Fi and Fj and

l=1,2,…,s, where s is the number of common edges in the

boundary between Fi and Fj.

If there be any strong pendant edge exists in the 0.67-fuzzy planar graph, self loop is considered for it. The edge membership value of the self loop is equal to the membership value of the pendant edge.

Example 7. In Figure 3, a 0.67 -fuzzy planar graph

),,(= EV where },,,{= dcbaV is given. For this

graph

0.9=)(0.8,=)(0.7,=)(0.6,=)( dcba Here

E={((a,b), 0.5), ((a,c), 0.4), ((a,d), 0.55), ((b,c), 0.45), ((b,c),

0.6), ((c,d, 0.7)}.

So the 0.67-fuzzy planar graph has the fuzzy faces F1 (bounded by ((a,b), 0.5), ((a,c), 0.4), ((b,c), 0.45)), F2

(bounded by ((a,d), 0.55), ((c,d), 0.7), ((a,c), 0.4)), F3 (bounded by ((b,c), 0.45), ((b,c), 0.6)) and outer fuzzy face F4 (surrounded by ((a,b), 0.5), ((b,c), 0.6), ((c,d), 0.7), ((a,d), 0.55).

The fuzzy dual graph is constructed as following. Here all the fuzzy faces are strong fuzzy faces. The vertex set

},,,{= 4321 xxxxV ' where the vertex ix is taken

corresponding to the strong fuzzy face Fi, i=1, 2,3, 4. So

σ'(x1)=max{0.5, 0.4, 0.45}=0.5, σ'(x2)=max{0.55, 0.7,

0.4}=0.7, σ'(x3)=max{0.45, 0.6}=0.6, and σ'(x4) = max{0.5,

0.6, 0.7, 0.55}=0.7.

,0.45=),(=),( 1131 cbxx

0.4=),(=),( 1121 caxx ,

0.5=),(=),( 1141 baxx ,

0.6=),(=),( 2143 cbxx ,

0.7=),(=),( 1142 dcxx ,

0.55=),(=),( 1242 daxx .

The edge set E’={((x1,x3), 0.45), ((x1, x2), 0.4), ((x1, x4), 0.5),

((x3, x4), 0.6), ((x2, x4), 0.7), ((x2, x4), 0.55)}.

The fuzzy dual graph ),,(= '''' EV is shown in

dotted line (see Fig. 4).

Weak edges in 0.67-planar graphs are not considered for any calculation in fuzzy dual graphs.

Fig. 4. Example of fuzzy dual graph

Theorem 8. Let ψ be a 0.67-fuzzy planar graph whose number

of vertices, number of fuzzy edges and number of strong faces

be n, p and m respectively.



Also let ψ’ be the fuzzy dual graph of ψ. Then (i) the number of

vertices of ψ’ is equal to m, (ii) number of edges of ψ’is equal

to p, (iii) number of fuzzy faces of ψ’ is equal to n.

VI. CONCLUSION

In this paper, we have introduced the fuzzy multigraphs, fuzzy planar graphs and fuzzy dual graphs. We also defined fuzzy planarity value and formula to find it for fuzzy complete graphs. Strong fuzzy planar graph is introduced here. A subclass of strong fuzzy planar graph viz. 0.67-fuzzy planar graph is defined. In near future we shall design a suitable algorithm to test fuzzy planarity.

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