Fuzzy graph and relation

Outline

• Graph and fuzzy graph• Characteristics of fuzzy relations• Types of fuzzy relations

Graph and fuzzy graph

• Graph

Graph and fuzzy graph

• Fuzzy graph~

V : is fuzzy node~

Graph and fuzzy graph

• Fuzzy graph and fuzzy relation

Graph and fuzzy graph

• α-cut of fuzzy graph

Graph and fuzzy graph

• α-cut of fuzzy graph

Graph and fuzzy graph

• Fuzzy network

Characteristics of fuzzy relations

• Reflexive relation

ive.antireflex 1),(,

e;irreflexiv 1),(,

xxAx

xxAx

R

R

Characteristics of fuzzy relations

• Symmetric relation

ric.antisymmetperfect

0),( 0),(,,,

ic;nonsymmetror asymmetric

),( ),(,,,

ric;antisymmet 0),( ),(or

),( ),(,,,

symmetric; ),( ),(,,

xyyxyxAAyx

xyyxyxAAyx

xyyx

xyyxyxAAyx

xyyxAAyx

RR

RR

RR

RR

RR

(

(

(

)

)

)

( )

Characteristics of fuzzy relations

• An example of perfect antisymmetric

Characteristics of fuzzy relations

• Transitive relation

),())],(),,([min(max),(

),(),,(),,(

zxzyyxzx

AAzxzyyx

RRRRy

R

Characteristics of fuzzy relations

• An example of transitive relation

Characteristics of fuzzy relations

• Transitive closure

𝑘≥𝑛

Characteristics of fuzzy relations

• An example of transitive closure

Types of fuzzy relation

• Fuzzy equivalence relationReflexive relationSymmetric relationTransitive relation

Types of fuzzy relationa b c d

a 1.0 0.8 0.4 0.1

b 0.8 1.0 0.0 0.0

c 0.4 0.0 1.0 0.5

d 0.1 0.0 0.5 1.0

(i) Reflexive relation √

(ii) Symmetric relation √

(iii) Transitive relation

a b c d

a 1.0 0.8 0.4 0.4

b 0.8 1.0 0.4 0.1

c 0.4 0.4 1.0 0.5

d 0.4 0.1 0.5 1.0

X

Types of fuzzy relation

• Fuzzy equivalence relation applications– Partition of sets

Types of fuzzy relation

• Fuzzy equivalence relation applications– Partition by α-cut

Types of fuzzy relation

• Fuzzy equivalence relation applications– Set similar to element x

Set similar to d

Set similar to e

Types of fuzzy relation

• Fuzzy compatibility relationReflexive relationSymmetric relation

Types of fuzzy relation

• Fuzzy pre-order relationReflexive relationTransitive relation

a b c

a 1.0 1.0 0.4

b 0.0 1.0 0.3

c 0.0 1.0 1.0

a b c

a 1.0 1.0 0.4

b 0 1.0 0.3

c 0 1.0 1.0

Types of fuzzy relation

• Fuzzy semi-pre-order or nonreflexive fuzzy pre-order relationX Reflexive relationTransitive relation

a b c

a 0.2 0.6 0.3

b 0.0 0.6 0.3

c 0.0 0.6 0.3

Types of fuzzy relation

• Anti-reflexive fuzzy pre-order relationAnti-reflexive relationTransitive relation

a b c

a 0.0 0.4 0.3

b 0.0 0.0 0.0

c 0.0 0.0 0.0

(x,x) = 0

Types of fuzzy relation

• Fuzzy order relationReflexive relationAntisymmetric relation

Transitive relation

• Perfect fuzzy order

, 𝑥≠ 𝑦

Types of fuzzy relation

• Fuzzy order relation crisp order

Types of fuzzy relation

• Fuzzy order relation Dominating class

Dominated class

Types of fuzzy relation

Types of fuzzy relation

• Dissimilitude relationAntireflexive relation

Symmetric relation

Min-Max transitive relation

Types of fuzzy relation

• ex

Min number = 0.2

Types of fuzzy relation

• exa b c d

a 0.0 0.4 0.1 0.0

b 0.4 0.0 0.2 1.0

c 0.1 0.2 0.0 0.3

d 0.0 1.0 0.3 0.0

(d,c) ; 0.0 v 0.1 = 0.11.0 v 0.2 = 1.00.3 v 0.0 = 0.30.0 v 0.3 = 0.3

Min number = 0.1

(b,c); 0.4 v 0.1 = 0.40.0 v 0.2 = 0.20.2 v 0.0 = 0.21.0 v 0.3 = 1.0

Min number = 0.2

Types of fuzzy relation

• Fuzzy morphism

Homomorphism

Fuzzy homomorphism

Types of fuzzy relation

• Fuzzy morphism

Strong homomorphism

Fuzzy strong homomorphism

Types of fuzzy relation

• Fuzzy homomorphism

Types of fuzzy relation

• Fuzzy strong homomorphism