COVERING IN OPERATIONS ON FUZZY GRAPHS

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1127

COVERING IN OPERATIONS ON FUZZY GRAPHS

Vinothkumar N

^†

, Ramya R

^∗

† Assistant professor, Department of Mathematics, Bannariamman Institute of Technology, Erode, Tamilnadu, India,

^∗ Assistant professor, Department of Mathematics, Bannariamman Institute of Technology, Erode, Tamilnadu, India,

Abstract

Let G be a fuzzy graph. A vertex cover of fuzzy graphs is a set of vertices such that each strong edge of the fuzzy graph is incident to at least one vertex of the set.In thispaperpresented the idea of covering in fuzzy graphs and define minimum covering in fuzzy graphs in thefirstpart and also make known tothe some operations in fuzzy graphs.

Further investigate the covering in fuzzy graphs operations give fuzzy examples to explain the results.

Keyword:. Fuzzy graphs, covering, covering number

1. Introduction

ThefirstdefinitionoffuzzygraphswasproposedbyKafmannfrom thefuzzyrelations introduced by Zadeh. Although Rosenfeld introduced another elaborated definition, includingfuzzyvertexandfuzzyedges,andseveralfuzzyanalogsofgraphtheoreticconceptssuch as paths, cycles, connectedness and etc. The concept of domination in fuzzy graphs was investigatedbyA.Somasundaram,S.SomasundaramandA.Somasundarampresentthe

conceptsofindependentdomination,totaldomination,connecteddominationoffuzzygraphs.C.

Natarajanand S.K. Ayyaswamy introduce the strong (weak) domination in fuzzy graph.

The first definition of intuitionistic fuzzy graphs was proposed by Atanassov. The concept of dominationinintuitionisticfuzzygraphs was investigated by R.parvathi and

G.Thamizhendhi.

In thispaperpresented the idea of covering in fuzzy graphs and define minimum covering in fuzzy graphs in thefirstpart and also make known tothe some operations in fuzzy graphs. Further investigate the covering in fuzzy graphs operations give fuzzy examples to explain the results.

2. Preliminaries

A fuzzy graph G(



,



)is a defined by the couple of fuzzy membership functions.

:V [0,1]



 _,



:VV [0,1]

wherefor allu,vV,every edges in G(



,



)fulfills the condition ).

( ) ( ) ,

(u v



u



v



 

The fuzzycardinality ofany subset

S  V

are defined to be the sum of membership

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1128

value of every elements in S,

i.e



S v

v)

 (

.

The order p of a fuzzy graph G(



,



) are defined to be the sum of membership values of vertex in G(



,



)

i.e



V x

x)

 (

The size q of a fuzzy graph G(



,



) are defined to be the sum of membership values of edges in G(



,



)

i.e



E uv

uv)

 (

.

Let G(



,



) be a fuzzy graph the degree of a vertex v to be







v u

uv v

d ( )  ( )

.

The minimum degree of G is

 ( G )  min{ d ( u ) / u  V }

and the maximum degree of G is

 ( G )  max{ d ( u ) / u  V }

.

An edge of a fuzzy graph is termed an effective edge if

( , ) u v min( ( ) u ( )). v

    

the neighborhood of u is defined by set

 

( ) / ( , ) min( ( ) ( ))

N u   v V  u v   u   v

and N[u] N(u){u} is called the

closed neighborhood of u.







) (

) ( )

(

u N v

N u v

d



is so-called the neighborhood degree of u.

The minimum neighborhood degree of G is



_N

( G )  min{ d

_N

( v ) / v  V }

and the maximum neighborhood degree of G is



_N

( G )  max{ d

_N

( u ) / u  V }

.

3. Covering in Fuzzy Graphs

The idea of covering in fuzzy graph was presented by Somasundaram. The author also defined node covering and arc covering in fuzzy graphs using effective arcs and scalar cardinality. According to the author, a node cover in a fuzzy graph G : (V, σ, µ) is a subset D of V such that for all effective arc e = (u, v), any one of u, v is in D. The minimum scalar cardinality of a node cover of G is called the covering number of G and is represented by αo(G) or αo. Further he defined the concept of coverings in a smaller domain of effective arcs and using scalar cardinality.

Definition 3.1

A subset D of V in a fuzzy graph G V( , , )

 

is said to be a cover of G V( , , )

 

such that every strong edges in G V( , , )

 

are incident with any one vertex in D.

The vertex cover number a fuzzy graph G V( , , )

 

is denoted by



⁰

( ) G

and is defined by minimum cardinality among all the minimal vertex cover of G V( , , )

 

_.

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1129

Theorem 3.1

A fuzzy graph G V( , , )

 

without isolated vertex then the covering number

0

( ) ( )

2 G O G

 

.

Proof: Let G V( , , )

 

be a fuzzy graph without isolated vertex and the D be the cover of G V( , , )

 

. D is a vertex cover then it covers all the strong edges in G V( , , )

 

. Therefore V-D also vertex cover of G V( , , )

 

_sinceG V( , , )

 

does not have an isolated vertex.

 

0

( ) min ,

( ) ( )

2 G D V D

G O G



 



Hence proved.

Example 3.1

G V( , , )

 

Figure 2.2.1

In the above example ,

 ^{ab bc de} ^, ^, 

are strong edges and the vertex cover of ( , , )

G V

 

_is

  ^{b e} ^,

and the vertex covering number of G V( , , )

 

_is



0

( ) 0.5 G 

. Definition 3.2

Let

G

¹

( ,  

¹ ¹

)

and

G

²

(  

²

,

²

)

be two fuzzy graphs on V1 and V2 respectively with

V

¹

  V

²



. The union of G1 and G2 is the fuzzy graph G on V¹V²

defined by

1 2 1 2 1 2

( ) ( , )

G  G  G       

where

1 1

1 2

2 2

( ) if u V

( )(u)

( ) u

u if u V

  



  

      

_and

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1130

1 1

1 2 2 2

( ) if u, v V

( )(uv) ( ) u, v V

0 uv

uv if otherwise



  

  

 

    

 

 

Theorem 3.2

The fuzzy graphs

G V

¹

( ,

¹

 

¹

,

¹

)

and

G V

²

( ,

²

 

²

,

²

)

with covering

1

D

2

D and

respectively. Then

(D

¹

 D )

²

is the minimum covering of

G

¹

 G

² . Proof: Let

G V

¹

( ,

¹

 

¹

,

¹

)

and

G V

²

( ,

²

 

²

,

²

)

fuzzy graphs with covering

1

D

2

D and

respectively. In

G

¹

 G

²

the edges of the forms

1 1

1 2 2 2

( ) if uv V

( )(uv) ( ) uv V

0 uv

uv if otherwise



  

  

 

    

 

 

therefore every strong edges in

G

¹

 G

²

covered by

(D

¹

 D )

²

.Hence proved.

Example 3.2

Figure 3.2

From the above example the covering set of the graphs

G V

¹

( ,

¹

 

¹

,

¹

)

and

2

( ,

2 2

,

2

) G V  

are

  ^{b c} ^,

_and

  ^e

respectively. The covering number of the graphs

1

( ,

1 1

,

1

) G V  

and

G V

²

( ,

²

 

²

,

²

)

are 0.5 and 0.3. The covering sets of the graph

1 2

G  G

are

 ^{b c e} ^{, ,} 

and the covering number is 0.8.

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1131

Definition 3.3

Let

G

¹

( ,  

¹ ¹

)

and

G

²

(  

²

,

²

)

is two fuzzy graphs on V1 and V2 respectively with

V

¹

  V

²



. The join of G1 and G2, denoted by

G

¹

 G

², is the fuzzy graph on

1 2

V  V

defined as follows.

1 2

(

1 2

,

1 2

)

G  G       

Where

1 1

1 2

2 2

( )( ) ( )

( )

u if u V

u u if u V

  



  

      

_And

1 1

1 2 2 2

1 2 1 2

( ) ,

( )( ) ( ) ,

( ) ( ) if u V uv if u v V uv uv if u v V

u v and v V



  

 

  

 

    

    

 

Theorem 3.3

The fuzzy graphs

G V

¹

( ,

¹

 

¹

,

¹

)

_and

G V

₂

( ,

₂

 

₂

,

₂

)

with covering

1

D

2

D and

respectively. Then

( V

¹

 D )

²

or

(D

¹

 V

²

)

1 2

G  G

.

Proof: Let

G V

¹

( ,

¹

 

¹

,

¹

)

and

G V

²

( ,

²

 

²

,

²

)

1

D

2

D and

respectively. In

G

¹

 G

²

1 1

1 2 2 2

1 2 1 2

( )

( )( ) ( )

( ) ( ) if u V uv if uv E uv uv if uv E

u v and v V



  

 

  

 

    

    

 

the strong edges in

G V

¹

( ,

¹

 

¹

,

¹

)

and

G V

²

( ,

²

 

²

,

²

)

are covered by

D and

¹

D

² respectively. Now we prove edges of the form

uv if u V and v V 

¹



²

is covered by

1 2

V orV

. The edges of the form

uv if u V and v V 

¹



²

are strong edges and incident with

1 2

V andV

. This implies

V andV

¹ ²

both covered the edges of the form. Therefore

1 2

( V  D )

or

(D

¹

 V

²

)

is a is the minimum covering of

G

¹

 G

²

.Hence proved.

Example 3.3

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1132

Figure 3.3

G V

¹

( ,

¹

 

¹

,

¹

)

and

2

( ,

2 2

,

2

) G V  

are

  ^{b c} ^,

_and

  ^e

respectively . the covering number of the graphs

1

( ,

1 1

,

1

) G V  

and

G V

²

( ,

²

 

²

,

²

)

G

¹

 G

² are

 ^{b c e f g} ^{, , , ,} 

_and

 ^{a b c d e} ^{, , , ,} 

Definition 3.4

Let

G

¹

 ( ,  

¹ ¹

)

and

G

²

(  

²

,

²

)

is two fuzzy graphs on V1 and V2 respectively.

Then the Direct product of G1 and G2, denoted by

G G

¹ ²

, is the fuzzy graph on

V V

¹



² defined as follows

1 2 1 2 1 2

1 2 1 2 1 1 2 2

1 2 1 2 1 2 1 1 1 2 2 2 1 1 1 2 2 2

( , )

( )( ) ( ) ( )

( )(( ), ( )) ( ) ( ) if ( ) ( )

G G where

u u u u and

u u v v u v u v u v E and u v E

   

   

   



 

   

Theorem 3.4

The Fuzzy graphs

G V

¹

( ,

¹

 

¹

,

¹

)

and

G V

²

( ,

²

 

²

,

²

)

with covering

1

D

2

D and

respectively. Then

( V

¹

 D )

²

or

(D

¹

 V

²

)

G G

¹ ² .Proof Let

G V

¹

( ,

¹

 

¹

,

¹

)

and

G V

²

( ,

²

 

²

,

²

)

1

D

2

D and

respectively. In

G G

¹ ²

1 2 1 2 1 2 1 1 1 2 2 2 1 1 1 2 2 2

(   )(( u u ), ( v v ))   ( u v )   ( u v ) if ( u v )  E and u v ( )  E

Suppose

( u v

^{1 1}

)  E and u v

¹

(

^{2 2}

)  E

²

are the strong edged in

G V

¹

( ,

¹

 

¹

,

¹

)

and

G V

²

( ,

²

 

²

,

²

)

therefore we get

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1133

   

1 2 1 2 1 2 1 1 1 2 2 2

1 1 1 1 2 2 2 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2

( )(( ), ( )) ( ) ( )

( ) (v ) ( ) (v )

( ) ( ) (v ) (v )

( )(( ), ( )) ( ) (v )

u u v v u v u v

u u

u u v v u u v

   

   

     

 

   

 

This implies the edge

(( u u

^{1 2}

), ( v v

^{1 2}

))

in

G G

¹ ²

is a strong edge. Now we prove these strong edges covered by

( V

¹

 D )

²

or

(D

¹

 V

²

)

. The strong edge

(( u u

^{1 2}

), ( v v

^{1 2}

))

in

G G

¹ ²

note that at least one of the vertex in

1 1 1

u or v  D

and

u or v

² ²

 D

²

.If

u

¹

 D and v

¹ ¹

 D

¹

and

u

²

 D and v

² ²

 D

² therefore

   

1 2 1 2 1 2 1 1 1 2 2 2

1 1 1 1 2 2 2 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2

( )(( ), ( )) ( ) ( )

( ) (v ) ( ) (v )

( ) ( ) (v ) (v )

( )(( ), ( )) ( ) (v )

u u v v u v u v

u u

u u v v u u v

   

   

     

 

   

 

This implies

( u u

^{1 2}

) D  

¹

V or (

²

v v

^{1 2}

)  D

²

. Hence the set

( V

¹

 D )

² or

1 2

(D  V )

is a covering set of

G G

¹ ² . Example 3.4

G V

¹

( ,

¹

 

¹

,

¹

)

and

2

( ,

2 2

,

2

) G V  

are

  ^{b c} ^,

_and

  ^g

respectively .covering number of the graphs

1

( ,

1 1

,

1

) G V  

and

G V

²

( ,

²

 

²

,

²

)

are 0.6 and 0.4.

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1134

Figure 3.4

G V

¹

( ,

¹

 

¹

,

¹

)

and

2

( ,

2 2

,

2

) G V  

are

  ^{b c} ^,

_and

  ^g

respectively. The Covering number of the graphs

1

( ,

1 1

,

1

) G V  

and

G V

²

( ,

²

 

²

,

²

)

are 0.6 and 0.4.The covering sets of the graph

G G

¹ ² are

 ⁽ ^ag ^{), (} ^bg ^{), (} ^cg ^{), (dg)} 

_and

 ^{(be), (} bf ), (b ), (ce), (cf), (cg) g 

the covering number is 1.4.

Definition 3.5

Let

G

¹

 ( ,  

¹ ¹

)

and

G

²

(  

²

,

²

)

is two fuzzy graphs on V1 and V2

respectively. Then the semi product of G1 and G2, denoted by

G

¹

G

²

, is the fuzzy graph

on

V V

¹



²

defined as follows

1 2 1 2 1 2

1 2 1 2 1 1 2 2

( , )

( )( ) ( ) ( )

G G where

u u u u

   

   



 

1 1 2 2 2 1 1 2 2 2

1 2 1 2 1 2

1 1 1 2 2 2 1 1 1 2 2 2

( ) ( ) ( )

( )((u , )( , ))

( ) ( ) if ( ) ( )

u u v if u v and u v E u v v

u v u v u v E and u v E

 

 

  

      

Theorem 3.6

The fuzzy graphs

G

¹

 (V ,

¹

 

¹

,

¹

)

and

G

²

(V ,

²

 

²

,

²

)

with covering sets D1 and D2 respectively. Then the set

( V

¹

 D

²

)

is the minimum covering set of

G

¹

G

² . Proof: Let

G V

¹

( ,

¹

 

¹

,

¹

)

and

G V

²

( ,

²

 

²

,

²

)

1

D

2

D and

respectively. In

G

¹

G

²

1 1 2 2 2 1 1 2 2 2

1 2 1 2 1 2

1 1 1 2 2 2 1 1 1 2 2 2

( ) ( ) ( )

( )((u , )( , ))

( ) ( ) if ( ) ( )

u u v if u v and u v E u v v

u v u v u v E and u v E

 

 

  

      

Clearly we note that the edges of the form i)

(( , u u

¹ ²

), ( , )) v v

¹ ²

if

u

¹

 v

¹

and

u v

^{2 2}

is a strong edge in

G

²

(V ,

²

 

²

,

²

)

. ii)

(( , u u

¹ ²

), ( , )) v v

¹ ²

if

u v

^{1 1}

and

u v

^{2 2}

is a strong edge in

G

¹

 (V ,

¹

 

¹

,

¹

)

and

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1135

2

( ,

2 2

,

2

) G V  

respectively.

These are the strong edges in

G

¹

G

²

. Now we prove these edges covered by the set

( V

¹

 D

²

)

.

Case (i) :

(( , u u

¹ ²

), ( , )) v v

¹ ²

if

u

¹

 v

¹

and

u v

^{2 2}

is a strong edge in

2

(V ,

2 2

,

2

)

G  

.

If

u

¹

 D

¹and suppose

u

²

 D and v

² ²

 D

²_since

D

₂is a minimum covering of

2

(V ,

2 2

,

2

)

G  

. Therefore

1 2 1 2 1 2 1 1 2 2 2

1 1 2 2 2 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2

( )(( , ), ( , )) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )(u )

u u v v u u v

u u v

u u u v

u u v

   

  

   

  

  

   

   

Note that the edges in this case and

u

¹

 D

¹

is covered by the vertex

1 2

(

1 2

)

u u  V  D

Case (ii) :

(( , u u

¹ ²

), ( , )) v v

¹ ²

if

u v

^{1 1}

and

u v

^{2 2}

is a strong edge in

1

(V ,

1 1

,

1

) G   

and

G V

²

( ,

²

 

²

,

²

)

respectively.

The strong edge

(( u u

^{1 2}

), ( v v

^{1 2}

))

in

G G

¹ ²

note that at least one of the vertex in

1 1 1

u or v  D

and

u or v

² ²

 D

²

.If

u

¹

 D and v

¹ ¹

 D

¹

and

u

²

 D and v

² ²

 D

² therefore

   

1 2 1 2 1 2 1 1 1 2 2 2

1 1 1 1 2 2 2 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2

( )(( ), ( )) ( ) ( )

( ) (v ) ( ) (v )

( ) ( ) (v ) (v )

( )(( ), ( )) ( ) (v )

u u v v u v u v

u u

u u v v u u v

   

   

     

 

   

 

This implies

V

¹

 D

²

cover the edges in case (ii). Hence the set

( V

¹

 D )

² is a covering set of

G

¹

G

²

. Therefore

V

¹

 D

²

is set covered the all strong edges in

G

¹

G

²

. Hence proved.

Example:3.6

(10)

1136

G V

¹

( ,

¹

 

¹

,

¹

)

and

2

( ,

2 2

,

2

) G V  

are

  ^{b c} ^,

_and

  ^g

1

( ,

1 1

,

1

)

G V  

_and

G V

₂

( ,

₂

 

₂

,

₂

)

are 0.6 and 0.4.

Figure 3.6 The covering sets of the graph

G G

¹ ²

are

 ⁽ ^ag ^{), (} ^bg ^{), (} ^cg ^{), (dg)} 

_{and the}

covering number is 1.4.

Definition 3.7

Let

G

¹

 ( ,  

¹ ¹

)

and

G

²

(  

²

,

²

)

respectively. Then the Cartesian product of G1 and G2, denoted by

G

¹

 G

², is the fuzzy graph on

V V

¹



²

defined as follows

 

 

























otherwise

v u if v u u

v v u u

and u u

u u

where G

G

0 ) ( ) (

) ( ) ( ))

, ( ), , )((

(

) ( ) ( ) , )(

(

) ,

(

2 2 1

1 1 2 2

1 1 2

2 2 1 1 2

1 2 1 2 1

2 2 1 1 2 1 2 1

2 1 2 1 2 1

















Theorem 3.7

The fuzzy graphs

G

¹

 (V ,

¹

 

¹

,

¹

)

and

G

²

(V ,

²

 

²

,

²

)

 D

1

 ( V

2

 D

2

)    ( V

1

 D

1

)  D

2



G

¹

 G

²_.

Proof: Let

G

¹

 (V ,

¹

 

¹

,

¹

)

and

G

²

(V ,

²

 

²

,

²

)

be the fuzzy graphs with covering set D1 and D2 respectively. In

G

¹

 G

²the edges of the form

1 1 2 2 2 1 1

1 2 1 2 1 2 2 2 1 1 1 2 2

( ) ( )

( )(( , ), ( , )) ( ) ( )

0 u u v if u v

u u v v u u v if u v

otherwise

 

   

 

 

    

 

Clearly we note that the edges of the form

(11)

1137

i)

(( , u u

¹ ²

), ( , )) v v

¹ ²

if

u

¹

 v

¹

and

u v

^{2 2}

is a strong edge in

G

²

(V ,

²

 

²

,

²

)

. ii)

(( , u u

¹ ²

), ( , )) v v

¹ ²

if

u

²

 v

²

and

u v

^{1 1}

is a strong edge in

G

¹

 (V ,

¹

 

¹

,

¹

)

. These are the strong edges in

G

¹

 G

² . Now we prove these edges covered by the set

 D

1

 ( V

2

 D

2

)    ( V

1

 D

1

)  D

2



. Case (i) :

(( , u u

¹ ²

), ( , )) v v

¹ ²

if

u

¹

 v

¹

and

u v

^{2 2}

is a strong edge in

2

(V ,

2 2

,

2

) G  

_.

If

u

¹

 D

¹

and suppose

u

²

 D and v

² ²

 D

²

since

D

²

is a minimum covering of

2

(V ,

2 2

,

2

)

G  

. Therefore

1 2 1 2 1 2 1 1 2 2 2

1 1 2 2 2 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2

( )(( , ), ( , )) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )(u )

u u v v u u v

u u v

u u u v

u u v

   

  

   

  

  

   

   

u

¹

 D

¹

 

1 2 1

(

2 2

)

u u  D  V  D

If

u

¹

 D

¹

and suppose

u

²

 D and v

² ²

 D

²

since

D

²

2

(V ,

2 2

,

2

)

G  

. Therefore

1 2 1 2 1 2 1 1 2 2 2

1 1 2 2 2 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2

( )(( , ), ( , )) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )(u )

u u v v u u v

u u v

u u u v

u u v

   

  

   

  

  

   

   

u

¹

 D

¹

 

1 2

(

1 1

)

2

u u  V  D  D

.

This implies edges in case (i) are covered by set

 D

1

 ( V

2

 D

2

)    ( V

1

 D

1

)  D

2



. Case (ii) :

(( , u u

¹ ²

), ( , )) v v

¹ ²

if

u

²

 v

²

and

u v

^{1 1}

is a strong edge in

1

(V ,

1 1

,

1

) G   

. If

u

²

 D

²

and suppose

u

¹

 D and v

¹ ¹

 D

¹

since

D

¹

1

(V ,

1 1

,

1

) G   

. Therefore

1 2 1 2 1 2 2 2 1 1 1

2 2 1 1 1 1

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2

( )(( , ), ( , )) ( ) ( )

( ) ( ) ( )

( ) ( ) (v ) (u )

( )( ) ( )(u )

u u v v u u v

u u v

u u

u u v

   

  

   

  

  

   

   

u

²

 D

²

 

1 2

(

1 1

)

2

u u  V  D  D

(12)

1138

If

u

¹

 D

¹

and suppose

u

²

 D and v

² ²

 D

²

since

D

²

2

(V ,

2 2

,

2

)

G  

. Therefore

1 2 1 2 1 2 1 1 2 2 2

1 1 2 2 2 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2

( )(( , ), ( , )) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )(u )

u u v v u u v

u u v

u u u v

u u v

   

  

   

  

  

   

   

u

¹

 D

¹

 

1 2

(

1 1

)

2

u v  V  D  D

.

This implies edges in case (ii) are covered by set

 D

1

 ( V

2

 D

2

)    ( V

1

 D

1

)  D

2



.

Therefore

 D

1

 ( V

2

 D

2

)    ( V

1

 D

1

)  D

2



is set covered the all strong edges in

G

¹

 G

². Hence proved.

Example 3.7

Figure 2.2.7

G V

¹

( ,

¹

 

¹

,

¹

)

and

2

( ,

2 2

,

2

) G V  

are

  ^{a d} ^,

_and

  ^f

1

( ,

1 1

,

1

) G V  

and

G V

²

( ,

²

 

²

,

²

)

2

1

G

G 

_are

 ⁽ ae ^{), (} ag ^{), (} bf ^{), (} cf ), (de), (dg) 

(13)

1139

Definition 3.8

Let

G

¹

 ( ,  

¹ ¹

)

and

G

²

(  

²

,

²

)

respectively. Then the composition of G1 and G2, denoted by

1 2 1 2 1 1 2 2

(   ) (u , u )   ( ) u   ( ) u

, is the fuzzy graph on

V V

¹



²

defined as follows

1 2

(

1 2

,

1 2

)

G G     

Where

(  

¹ ²

)(u ,

¹

u

²

)  

¹

( ) u

¹

 

²

( ) u

²

and

1 1 2 2 2 1 1 2 2

1 2 1 2 1 2 2 2 1 1 1 1 1 2 2

2 2 2 2 1 1 1

( ) ( , )

( )((u , )( , )) ( ) ( , )

( ) ( ) ( , )

u u v if u v and u v u v v u u v if u v and u v u v u v otherwise

 

   

  

  

 

    

  



Theorem 3.7

The fuzzy graphs

G

¹

 (V ,

¹

 

¹

,

¹

)

and

G

²

(V ,

²

 

²

,

²

)

( D V

¹



²

)  ( V

¹

 D

²

)

1 2

G G

.

Proof Let

G

¹

 (V ,

¹

 

¹

,

¹

)

and

G

²

(V ,

²

 

²

,

²

)

be the fuzzy graphs with covering set D1 and D2 respectively. In

G G

¹ ²

the edges of the form

1 1 2 2 2 1 1 2 2

1 2 1 2 1 2 2 2 1 1 1 1 1 2 2

2 2 2 2 1 1 1

( ) ( , )

( )((u , )( , )) ( ) ( , )

( ) ( ) ( , )

u u v if u v and u v u v v u u v if u v and u v u v u v otherwise

 

   

  

  



   

  

 Case(i): In

((u

¹

u

²

)( v v

^{1 2}

))

,

u

¹

 v

¹

and

(u

²

v

²

)  E

² If

(u

²

v

²

)  E

²

be a strong edge in

G

²

(V ,

²

 

²

,

²

)

therefore

u or v

² ²

 D

² since

D

²

is coving of

G

²

. This implies

1 2 1 2 1 2 1 1 2 2 2

1 1 2 2 2 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2

( )((u )( )) ( ) ( , )

( ) ( ) (v )

( ) ( ) ( ) (v )

( )(u ) ( )( )

u v v u u v

u u

u u u

u v v

   

  

   

 

  

   

 

The edge

((u

¹

u

²

)( v v

^{1 2}

))

is strong edge in

G G

¹ ²

, the end vertices

1 2 1 2

(u u ) and ( v v )

. Any one of the end vertex in

( V

¹

 D

²

)

since

u or v

² ²

 D

² . Therefore these strong edges covered by

( V

¹

 D

²

)

. Case(ii): In

((u

¹

u

²

)( v v

^{1 2}

))

,

u

²

 v

²

and

(u

¹

v

¹

)  E

¹ If

(u

^{1 1}

v )  E

¹

be a strong edge in

G

¹

 (V ,

¹

 

¹

,

¹

)

therefore

u or v

¹ ¹

 D

¹ since

D

1

is coving of

G

¹

. This implies

(14)

1140

1 2 1 2 1 2 1 1 1 2 2

1 1 1 1 2 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2

( )((u )( )) ( ) ( )

( ) (v ) (u )

( ) (u ) (v ) (u )

( )(u ) ( )( )

u v v u v u

u u

u v v

   

  

   

 

  

   

 

The edge

((u

¹

u

²

)( v v

^{1 2}

))

is strong edge in

G G

¹ ²

, the end vertices

1 2 1 2

(u u ) and ( v v )

(D

¹

 V

²

)

since

u or v

¹ ¹

 D

¹

. Therefore these strong edges covered by

(D

¹

 V

²

)

. Case(iii): In

((u

¹

u

²

)( v v

^{1 2}

))

,

u

²

 v

²

and

(u

¹

v

¹

)  E

¹ If

(u

^{1 1}

v )  E

¹

be a strong edge in

G

¹

 (V ,

¹

 

¹

,

¹

)

therefore

u or v

¹ ¹

 D

¹ since

D

1is coving of

G

¹. This implies

1 2 1 2 1 2 2 2 2 2 1 1 1

1 1 1 1 2 2 2 2

1 2 1 2 1 2 1 2

( )((u )( )) ( ) ( ) ( )

( ) (v ) (u ) ( )

( )(u ) ( )( )

u v v u v u v

u v

u v v

    

   

  

   

 

The edge

((u

¹

u

²

)( v v

^{1 2}

))

is strong edge in

G G

¹ ²

, the end vertices

1 2 1 2

(u u ) and ( v v )

(D

¹

 V

²

)

since

u or v

¹ ¹

 D

¹

. Therefore these strong edges covered by

(D

¹

 V

²

)

. Hence the graph

G G

¹ ²

is covered by the set

( D V

¹



²

)  ( V

¹

 D

²

)

.

Example 3.8