Estimation of Isomorphism Degree of Fuzzy Graphs - EUSFLAT

Estimation of Isomorphism Degree of Fuzzy Graphs
Bershtein L.S.
Department of Informatics
Taganrog State University of Radioengineering
Nekrasovsky Street 44
347900 Taganrog, Russia
Tel.: +78634383289, Fax: +78634315300
e-mail: lsb@pbox.ttn.ru
Abstract
The questions of isomorphism degree
estimation of fuzzy graphs are discussed in
this paper. The definition of independent
fuzzy set of fuzzy graph is presented. The
estimation method of isomorphism degree
of fuzzy graphs is suggested and proved.
The example of isomorphism degree
estimation is considered.
Keywords: Fuzzy Graph, Isomorphism Degree,
Independent Degree, Fuzzy Independent Vertex
Set, Independent Fuzzy Set.
1 Introduction
A graph isomorphism search is important problem of
graph theory. It consists to define of bijective
correspondence existence which preserve adjacent
relation between vertex sets of two graphs [7]. In a
case of fuzzy graphs the notion of isomorphism is
fuzzy.
a
We consider fuzzy graphs [5]
a
* ; 8 and
a a
* < 8 , where X and Y are sets of vertices
a
and 8 ; ^ ¬ ; [ L [ M [ L [ M > _ [ L [ M ∈ ; ` and
a
8 ^ ¬ \ \ \ \ > _ \ \ ∈<
` are
fuzzy sets of edges with membership functions
µX:X 2 →[0,1] and µY:Y 2 →[0,1]. Let numbers of
vertices coincide (|X|=|Y|=n).
Let Φ ={F:X→Y} be a set of all bijective
correspondences which may be define on the set
X×Y.
Bozhenyuk A.V.
Department of Informatics
Taganrog State University of Radioengineering
Nekrasovsky Street 44
347900 Taganrog, Russia
Tel.: +78634310866, Fax: +78634315300
avb@pbox.infotecstt.ru
Definition 1 [3]. A value
PD[ ¬ [ [ ↔ ¬ \ \
I ; L M < L M
∀)
∈Φ
L = Q M = Q
is called an isomorphism degree of fuzzy graphs
* ;
a and * <
a .
A task of search degree isomorphism is NPcomplete
task as since |Φ|=n!
Here & is conjunction operation, which is defined as
a&b=min{a,b}. Operation ↔ is equivalence
operation, which is defined as
D ↔ E = D → E E → D . Operation →
is implication operation, which has property 0→b=1
(falsity implies anything) [4]. In Lukasiewicz logic it
is defined as D → E = PLQ^ − D + E`.
Crisp graphs may be consider as fuzzy graphs which
have membership functions equal 0 or 1.
Consequently, their isomorphism degree f may be 0
or 1 also. If f=1, then crisp graphs are isomorphous.
If f=0, then crisp graphs aren’t isomorphous.
Crisp graphs are defined some invariants:
independent number, dominating vertex number,
chromatic number and others [6]. If crisp graphs are
isomorphous (f=1), then their invariants coincide. If
invariants don’t coincide, then crisp graphs aren’t
isomorphous (f=1)
When we deal with fuzzy graphs, their invariants are
fuzzy values also. A task of fuzzy invariants
influence on degree isomorphism of fuzzy graphs
arises in this case. A correlation between
isomorphism degree and independent fuzzy sets of
fuzzy graphs is considered in this paper.

2 Isomorphism degree estimation
Consider a fuzzy subgraph a
* ;
a
8
, where
Xi⊆X, |Xi|=k ( N = Q ).
Definition 2 [1]. A value = − PD[ µ [ [
α
∈
is called an independence degree of fuzzy subgraph
* ;
a
.
Definition 3 [1]. A subset of vertices XK⊆X of fuzzy
graph * ;
a is called a maximal fuzzy independent
vertex set with degree α ; , if the condition
; ; α α ′ < is true for all X’⊃XK.
We define as α = PD[^α
α α `,
where
α
α
α are the degrees of maximal fuzzy
independent vertex sets, which have k elements. A
PD[
volume α means that fuzzy graph
a a
;
* ; 8
includes a fuzzy subgraph with k vertices and with
PD[
independent degree α and it doesn’t include any
;
subgraphs more than k vertices and with independent
PD[
degree more than α .
;
Definition 4. A fuzzy set
a
$ = ^<
> < > < Q > `
α α α is called
an independent fuzzy set of fuzzy graph *
a .
Property 1. The following proposition is true:
≥ ≥ ≥ ≥ ≥
α α α .
PD[
Volume α equals 1 if set X doesn’t contain any
;
PD[
vertices with loop, and volume α equals 0 if set
;
8 ;
a contains an edge with membership function
equals 1.
We consider a some bijective correspondence
) ; → < between vertex sets X and Y. Let f be
an isomorphism degree of fuzzy graphs
a a a a
* ; 8 and * < 8 . Let fk be an
isomorphism degree of fuzzy subgraphs
a a
* ; 8 and a a
* < 8 .
Property 2. The following proposition is true:
N Q IN
I ≥ = ∀ .
Proof. We renumber vertices of sets X and Y. We
mark vertices of subsets XK and YK as 1,2,…,k and
vertices of subsets X\XK and Y\YK as k+1, k+2,…,n.
Then the degree of isomorphism may be write as:
I
=
L=
N M = N
L=
Q M = N + Q
I
N
L=
Q M = Q
¬
¬
L=
N + Q M = Q
L=
Q M = N + Q
L=
N + Q M = Q
¬
;
;
;
¬
¬
[
[
;
;
[
L
L
L
[
[
¬
[
[
Property 2 is proved.
Property 3. Let
;
L
;
[
L
M
M
M
[
[
[
↔
↔
α and
M
L
M
↔
[
M
¬
¬
↔
↔
¬
<
<
<
¬
¬
\
\
<
↔
degrees of fuzzy subgraphs
following proposition is true:
∀N = Q α ; ↔ α < ≥ I N (1)
Proof. If α and α are an independence
;
<
degrees, then there are some vertices . ; [ [ ∈ ,
for which α ; = − µ ; [ [ , and there are
some \ \ ∈ < . , for which α< = − µ < \ \ .
This case is presented in Figure1.
<
\
L
L
L
\
\
\
¬
\
\
L
<
L
M
M
M
\
\
\
M
L
M
\
=
M
=
≤
α be an independence
<
* a and * <
a . Then
Figure 1: Example of case α = − µ [ [
;
and α = − µ \ \
<
We consider two cases.
Case 1. Vertex y1 corresponds vertex x1, (F(x1)=y1),
vertex y2 corresponds vertex x2 (F(x2)=y2). Then the
volume I may be estimate as:
I
=
a
* ;
N
≤
N
µ
−α
;
;
[
↔
[
− α
<
<
↔ µ
<
\
= α
;
\
;
=
↔ α
Case 2. Vertex \ corresponds vertex x1
( ) [ = \ ′ ), vertex \
corresponds vertex x2
( ) [ = \ ′ ), but \ ′ ≠ \ and (or) \ ′ ≠ \ .
; ? ;
;
< ? <
µ [ [
µ \ \
[
[
\
\
<
I
N
<
a
*

Then the expression
true. Hence we have
−
This case is presented in Figure 2.
a
* ;
µ \ \ ≥ µ \ ′ \ ′ is
W = µ \ ′ \ ′ ≥ α .
Figure 2: Example of case \ ′ ≠ \ and \ ′ ≠ \
We consider two cases again.
Case 2.1. The following proposition is true
<
α ↔ α ≥ α ↔ W . Then volume
;
may be estimate as:
I
=
N
≤
= α
;
N
<
µ
−α
;
;
N
[
↔α
<
N
;
[
↔
↔µ
−W
<
<
\ ′
= α
;
N
\ ′
↔W
Case 2.2. The following proposition is true
α ; ↔ α < < α ;
<
↔ W . (2)
Let vertices \ \ ∈ < correspond vertices
−
−
[ ′ [ ′ ∈ ; , ( ) \ = [ ′ and ) \ = [ ′ ).
Then µ [ [ ≥ µ [ ′ [ ′ is true and we
may write
W = − µ [ ′ [ ′ ≥ α . (3)
Proposition (2) involves α ; ↔ α <
;
> W ↔ α <
by condition (3). In this case volume I may be
N
estimate as:
I ≤ µ [ ′ [ ′ ↔ µ \ \ =
=
µ
N
= W
[ [
;
− W
; ? ;
[
[µ
[
µ [ µ [ µ
;
[µ
;
;
↔ α
↔
<
< α
−α
;
<
<
=
↔ α
< ? <
The property 3 is proved for any N = Q .
a
Let $ = ^<
> < > < Q>
`
α α α and
a
$ = ^<
> < > < Q > `
α α α
be
* a
independent fuzzy sets of fuzzy graphs * ;
a and <
\µ
\
<
µ
µ
\ \ µ
µ
\µ
<
\ \
\
=
<
=
a
*
I
N
correspondingly, and f be an isomorphism degree of
these fuzzy graphs.
Property 4. The following proposition is true:
I ≤ α ↔ α . (4)
=
Proof. Let a a
* ; 8 and a a
* < 8 be
some fuzzy subgraphs with independence degrees
α and
α correspondingly. We consider two
cases.
Case 1. Subset YK corresponds to subset XK
( ) ; = < ). This case is presented in Figure3.
a
* ;
Figure 3: Example of case
) ; = <
I ≤ I ≤ α ↔ α on
Then we can write
the base of properties 2 and 3.
Case 2. Subset YK doesn’t correspond to subset XK
( ) ;
Figure4.
= < ′ ≠ < ). This case is presented in
a
* ;
a
* ;
a
* µ
;
a
* ;
) ; N < N
) ; N < N µ
µ
) ; N < N
Figure 4: Example of case
a
*
a
*
a
* µ
) ; = < ′ ≠ <
Then independence degree α ′ of fuzzy subgraph
a a
* ′ < ′ 8 ′ may be estimate as α ′ ≤ α . We
also consider two cases.
Case 2.1. The following proposition is true
PD[ PD[ PD[
α ; ↔ α<
≥ α
; ↔ α′
< . Then
isomorphism degree may be estimate as:
a
*
a
*

I ≤ I ≤ α ↔ α ≤ α ↔ α
.
Case 2.2. The following proposition is true
PD[ PD[ PD[
α ↔ α < α ↔ α ′ . (5)
;
<
;
′
Let subset ; corresponds to subset <
( ) ; ′ = < ). Let α ′ be an independence degree
′
of subset ;
. Then we write
α ′
≤ α . (6)
Proposition (5) involves inequality
PD[
α ;
PD[
↔ α < < α ′ ;
PD[
↔ α <
by condition (6).
In this case isomorphism degree may be estimate as:
I ≤ I ≤ α ′
↔ α ≤ α ↔ α .
Property 4 is proved because we consider all cases.
Example. Estimate of isomorphism degree of the
fuzzy graphs, shown in the Figure 5.
[ [
\
[ \
[ \
Figure 5: Example of fuzzy graphs
′
<
*
a and
Independent fuzzy sets of these fuzzy graphs are
a
defined as $ = ^<
> < > < > < > `
and
a
$ = ^<
> < > < > < > `.
Then isomorphism degree is estimated as:
PD[ PD[
I α ↔ α =
≤
N =
= PLQ
;
↔
↔
↔
↔ =
Hence, we may assert that isomorphism degree of
these fuzzy graphs will not be larger then 0,8.
3 Conclusion
Proven proposition (4) allows estimate a possible
isomorphism degree of fuzzy graphs by fuzzy
independent sets. It is necessary to remark that the
estimate of isomorphism degree may be receive by
another invariants of fuzzy graphs, such as
dominating vertex sets, fuzzy graph kernels [1],
fuzzy chromatic set [2].
<
* a
\
References
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