Ji2416271633

N.Sathyaseelan, Dr.E.Chandrasekaran / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 4, July-August 2012, pp.1627-1633
SELF WEAK COMPLEMENTARY FUZZY GRAPHS
N.Sathyaseelan ∗ , Dr.E.Chandrasekaran∗∗
∗(Assistant Professor in Mathematics, T.K Government Arts College, Vriddhachalam – 606 001.)
∗∗(Associate Professor in Mathematics, Presidency College, Chennai. Tamilnadu, India.)

ABSTRACT  
In this paper, the order, size and degree of the G : (σ,  ) where σc = σ and µc(x,y) = σ(x) σ(y) -
nodes of the isomorphic fuzzy graphs are µ(x,y) x,y S.
discussed. Isomorphism between fuzzy graphs is
proved to be an equivalence relation. Weak III.ISOMORPHISM IN FUZZY GRAPHS
Isomorphism between fuzzy graphs is proved Definition: 3.1[2]
the partial order relation. Some properties of A homomorphism of fuzzy graph h :
self complementary and self weak G→G' is a map h : S→S' which satisfies,
complementary fuzzy graphs are discussed.
σ(x) ≤ σ'[ h(x) ] x S and
Keywords - Fuzzy relation, equivalence relation,
weak isomorphism, Self complementary fuzzy µ(x ,y) ≤ µ'[ h(x), h(y) ] x S
graphs, Self weak complementary fuzzy graphs. Definition: 3.2[2]
A isomorphism h : G→G' is a map h :
I. INTRODUCTION S→S' which is bijective that satisfies,
The concept of fuzzy relations which has σ(x) = σ' [ h(x) ] x S
a widespread application in pattern recognition was
µ(x,y) = G' [ h(x), h(y) ] x S
introduced by Zadeh in his classical paper in 1965.
P. Bhattacharya in [1] showed that a fuzzy graph Isomorphism between fuzzy graphs
can be associated with a fuzzy group in a natural denote as G G'.
way as an automorphism group. K.R.Bhutani in Remark : 3.3
[2] introduced the concept of weak isomorphism 1. A weak isomorphism preserves the weights of
and isomorphism between fuzzy graphs. the nodes but not necessarily the weights of the
edges.
II.PRELIMINARIES 2. An isomorphism preserves the weights of the
A fuzzy graph with S as the underlying set edges and the weights of the nodes.
is a pair G : (σ ,μ ) where σ : S →[0,1] is a fuzzy 3. An endomorphism of a fuzzy graph G: (σ , µ ) is
subset, μ : S × S →[0,1] is a fuzzy relation on a homomorphism of G to itself.
the fuzzy subset σ , such that μ (x, y) ≤ σ (x)∧ σ (y) 4. An automorphism of a fuzzy graph G: (σ , µ ) is
for all x,y ∈S . an isomorphism of G to itself
Throughout this paper S is taken as a finite set.sup 5. When the two fuzzy graphs G & G' are same the
p(σ)={u / σ(u) > 0},andsup p(μ )={(u,v) /μ (u,v) > weak isomorphism between them becomes an
0}. For the definitions that follow let G : (σ ,μ ) isomorphism .
and G' : (σ' ,μ' ), be the fuzzy graphs with
underlying sets S and S' respectively. In crisp graph, when two graphs are isomorphic
Definition :2.1[4] they are of same size and order. The following
theorem is analogous to this.
Given a fuzzy graph G : (σ ,μ ) with the underlying Theorem :3.4
set S, the order of G is defined and denoted as p
For any two isomorphic fuzzy graph their
= (x) and size of G is defined and denoted as
order and size are same.
q= x, y)
Definition :2.2 Proof:Let G: (σ, µ) and G': (σ', µ') are any two
A fuzzy graph G : (σ,µ) is connected if fuzzy graph with the underlying set S and S'
µ∞(x, y) > 0 x,y σ*. Where σ* = sup (σ) = { x respectively. If h: G→ G' is an isomorphism
S / σ(x) > 0 }. between the fuzzy graphs G and G' then it satisfies,

µ (x) = σ'[ h(x) ] x S
Definition :2.3.[3]
µ(x, y) = µ'[ h(x), h(y) ] x S
The complement of a fuzzy graph G : (σ,µ) is
also a fuzzy graph is denoted as Gc :(σ , µc) or (i).Order(G)= = Order(
G')

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(ii).Size(G)= = σ(x)=σ'[h(x)] x S
= Size(G') (1) µ(x,y)=µ'[h(x),h(y)] x S
(2)
Therefore for any two isomorphic fuzzy graph their µ∞(x, y) = sup { µk(x, y) / k = 1, 2, 3,… }
order and size are same.
= sup { xi-1, xi) /x = x0,x1,x2,.. xk-1,xk =y
Corollary:3.5 Converse of the result need not be
S, k = 1, 2,…}
true.
From equation (2),
Example: 3.6
= sup{ xi-1), h(xi)] / x = x0, x1, x2,…
Consider the fuzzy graph G and G' with
the underlying sets S = {a,b,c,d} and S' =
{a',b',c',d'} σ(x) = 1 x S, µ(a,b) = 0.25, xk-1, xk = y S, k = 1, 2,…}
µ(b,c) = 0.5, µ(c,d) = 0.25, (x) = 1 x S,
= sup{ µ'k [h(x), h(y)] / k=1, 2,…}
µ'(a',b') = 0.25, µ(b',c') = 0.5, µ(c',d') = 0.25. (refer
fig. 3.1) µ∞(x,y)=µ’∞[h(x),h(y)]
(3)
From this two graphs,
Therefore G is isomorphic to
Order (G) = p = = (a)+ (b)+ (c)+ (d)
=1+1+1+1 = 4 = Order(G') G' µ∞(x, y) = µ'∞[h(x), h(y)] x S

Size(G) = q = = G is connected iff µ∞(x, y) > 0 iff µ'∞[h(x), h(y)] >
µ(a,b)+µ(b,c)+µ(c,d) = .25+.25+.5= 1 =Size(G') 0 [by equation (3)]

Theorem: 3.7 iff G’ is connected. Hence the proof.

If G and G’ are isomorphic fuzzy graphs Theorem: 3.10
then the degree of their nodes are preserved.
Isomorphism between fuzzy graphs is an
Proof equivalence relation.

Let h : S→S’ be an isomorphism of G onto G. by Proof:
the definition (3.2),
Let G : (σ,µ), G': (σ',μ ') and G'' : (σ'',µ'') be
∑µ(x,y) = ∑µ' (h(x),h(y)) x S the fuzzy graph with the underlying set S, S' and
S'' respectively. Isomorphism between fuzzy graphs
d(u) = = = is an equivalence relation if it satisfies Reflexive ,
Symmetric and Transitive.
d(h(u))
Reflexive
Therefore for any two isomorphic graphs their
degrees of vertex are same. Consider the identity map h : S→S such
that h(x)=x x S
corollary:3.8 But the converse of the result need
not be true. Since the fuzzy graphs are isomorphic then h is a
bijective map satisfying,

σ(x) = σ[ h(x) ] x S
Theorem: 3.9

If G is isomorphic to G' then G is µ(x,y) = µ[ h(x), h(y) ] x S
connected iff G' is also connected.
Hence h is an isomorphism of the fuzzy graph to
Proof itself. Therefore G is isomorphic to itself.
Symmetric
Assume that G is isomorphic to G'. Let h : S→S' be an isomorphism of G onto
There exist a bijective map h : S→S' be an G’ then h is a bijective map such that
isomorphism of G onto G' such that h(x) = x', x S,
Satisfying

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h(x) = x', x S Example:4.3(referFig4.1)
(4)
Satisfying σ(x) = σ' [ h(x) ] x S The above two fuzzy graphs are of order 3 but
they are not weak isomorphic.
(5)
µ(x,y) = µ' [ h(x), h(y) ] x S Example : 4.4
(6) Let G : (σ ,μ ) and G' : (σ' ,μ' ), be the fuzzy
As h is bijective, by equation (4) h-1(x') = x, x S' graphs with underlying sets S = {a,b.c} and S' =
Using equation (5), σ[h-1(x')] = σ' (x') x S (7) {a',b,c'} where σ : S→[0,1], μ:S×S→[0,1] , σ' :
S'→[0,1], μ':S'×S'→[0,1] are defined as σ(a) = 1/2,
µ[h-1(x'), h-1(y')] = µ' (x',y') x S' σ(b) = 1/4, σ(c) = 1/3 ; µ(a,b) = 1/5, µ(b,c) = 1/5
µ(a,c) =1/4; σ' (a') =1/2, σ' (b') =1/4, σ' (c') = 1/3 ;
(8)
µ'(a',b') = 1/4, µ' (b',c') = 1/4 µ' (a',c') =1/4
Hence we get one-to-one, onto map
Definition h : S →S' as h(a) = a', h(b) = b' , h(c) =
h-1 : S'→S, which is an isomorphism from G' to G.
c' this h is a bijective mapping satisfying
i.e G G' G' G.
σ(x) = σ' [ h(x) ] x S
Transitive
µ(x,y) ≤ µ' [ h(x), h(y) ] x S
Let h : S→S' and g : S'→S'' be an
isomorphism of the fuzzy graphs G onto G' and G' (refer Fig.4.2)
onto G'' respectively. Theorem:4.5
Thereforeg h : S''→S such that (g h)x g(h(x))
Weak isomorphism between fuzzy graphs satisfies
x S the partial order relation.
Since h is a bijective map that satisfies h(x) = x',
x S. Proof
σ(x) = σ'[h(x)]andµ(x,y) = µ' [ h(x), h(y) ]
Let G : (σ,µ), G' : (σ',µ') and G'' : (σ'',µ'') be the
x S
fuzzy graph with the underlying set S,S' and S''
i.e., σ(x) = σ' (x') and µ(x,y) = µ' (x',y') x S respectively. Weak isomorphism between fuzzy
(9) graphs is partial order relation if it satisfies
Since g is a bijective map that satisfies Reflexive , Anti-Symmetric and Transitive.
g(x') = x'', x' S'.
σ'(x') = σ'' [ g(x') ] and Reflexive
µ'(x',y')=µ''[g(x'),g(y')] x' S'
Consider the identity map h : S→S such that
(10)
h(x)=x x S
From (4), (9) and (10)
σ(x) = σ' (x') = σ'' [g(x')] x' S'
Since the fuzzy graphs are isomorphic then h is a
bijective map satisfying,
= σ'' [g (h(x))] x S
σ(x) = σ'' [g h(x)] x S σ(x) = σ[ h(x) ] x S
µ(x,y) = µ' (x',y') = µ'' [g(x'), g(y')]
= µ'' [g(h(x)), g(h(y))] x S µ(x,y) = µ[ h(x), h(y) ] x S
µ(x,y) = µ'' [(g h)x, (g h)y] x S. Hence h is a weak isomorphism of the fuzzy graph
Therefore the transitive relation g h is an to itself. Therefore G is weak isomorphic to itself.
Anti – Symmetric
isomorphism between G and G''.
Let h be a weak isomorphism between S and S’ and
Hence isomorphism between fuzzy graphs is an g be a weak isomorphism between S' and S.
equivalence relation. That is h : S→S' is a bijective map, h(x) = x', x S
satisfying
IV. WEAK ISOMORPHISM IN FUZZY σ(x) = σ' [ h(x) ] x S
GRAPHS (11)
Definition: 4.1[2] µ(x,y) ≤ µ' [ h(x), h(y) ] x S
A weak isomorphisam h : G → G' is a (12)
map h : S →S' which is a bijective homomorphism g : S'→S is a bijective map, g(x') = x, x' S'
that satisfies,
satisfying
σ(x) = σ' [ h(x) ] x S
σ' (x') = σ[ g(x') ] x S'
µ(x,y) ≤ µ' [ h(x), h(y) ] x S (13)
Remark : 4.2 If the fuzzy graphs are weak µ' (x',y') ≤ µ[ g(x'), g(y') ] x S'
isomorphic then their order are same. But the fuzzy
(14)
graphs of same order need not be weak isomorphic.

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The inequalities (12) and (14) hold good on the That is, Gc is isomorphic to Gc'
finite sets S and S' only when G and G' have the Hence G and G' are isomorphic then Gc and Gc' are
same number of edges and the corresponding edges also isomorphic.
have same weight. Hence G and G' are identical. Sufficient Part:
Transitive Assume that Gc and Gc' are isomorphic then we
Let h : S→S' and g : S'→S'' be a weak isomorphism have to prove G and G' are isomorphic fuzzy
of the fuzzy graphs G onto G' and G' onto G'' graphs.
respectively. By our assumption there exist a bijective map
Therefore g h:S''→S such that (g h)x =g(h(x)) g : S→S' satisfying,
x S σ(x) = σ' [ g(x) ] x S
As h is a weak isomorphism, h(x) = x', x S (20)
satisfying µc(x,y) = µc' [ g(x), g(y) ] x S
σ(x) = σ' [ h(x) ] (21)
µ(x,y) ≤ µ' [ h(x), h(y) ] x S By the definition (2.3),
(15) µc(x,y) = σ(x) σ(y) - µ(x,y) x,y S
As g is a weak isomorphism, g(x') = x'', x' S' (22)
satisfying µc' [g(x),g(y)] = σ'[g(x)] σ' [ g(y) ]- µ' [ g(x), g(y)
σ'(x') = σ'' [ g(x') ] ]
µ' (x',y') ≤ µ'' [ g(x'), g(y') ] x' S'' From (20),µc'[g(x),g(y)] =σ(x) σ(y) - µ' [ g(x),
(16) From (15) and (16) g(y) ]
σ(x) = σ[h(x)] = σ' (x') = σ''[g(x')] x' S' From (22), µc(x,y) = σ(x) σ(y) - µ' [ g(x), g(y) ]
= σ'' [g (h(x))] x S That is, µ' [ g(x), g(y) ] = σ(x) σ(y) - µc(x,y)
σ(x) = σ'' [g h(x)] x S From (22), µ' [ g(x), g(y) ] = µ(x,y) x,y S
µ(x,y) ≤ µ[h(x), h(y)] = µ' (x',y') = µ'' [g(x'), g(y')] (23)
= µ'' [g(h(x)), g(h(y))] Hence from equation (20) and (23) g : S→S' is an
µ(x,y) ≤ µ'' [(g h)x, (g h)y] x S isomorphism between G and G'.
Therefore the transitive relation g h is a weak Hence Gc and Gc' are isomorphic then G and G' are
also isomorphic.
isomorphism between G and G''. Theorem :4.7
Hence weak isomorphism between fuzzy graphs is If there is a weak isomorphism between G and G'
a partial order relation. then there is a weak isomorphism between G c and
Gc'. where G : (σ,µ), G': (σ',µ') are any two fuzzy
Theorem :4.6
graphs with the underlying set S and S'.
Any two fuzzy graphs are isomorphic if and only if
Proof:
their complements are isomorphic.
Assume that G is a weak isomorphism between G
Proof:
and G'.
Necessary Part:
There exist a bijective map h : S→S' such that
Let G : (σ,µ) and G': (σ',µ') be the two given fuzzy
h(x) = x', x S satisfying,
graphs.

As G is isomorphic to G', there exist a bijective σ(x) = σ' [ h(x) ] x S
map h : S →S' such that h(x) = x', x S satisfying, (24)
µ(x,y) ≤ µ' [ h(x), h(y) ] x S
σ(x) = σ' [ h(x) ] x S (25)
(17) As h-1 : S'→S is also bijective for x' S' such that
µ(x,y) = µ' [ h(x), h(y) ] x S h-1(x') = x.
(18) From (24), σ [h-1 (x')] = σ' (x') x S'
By definition (2.3), (26)
µc(x,y) = σ(x) σ(y) - µ(x,y) x,y S By the definition of (2.3),
(19) µc(x,y) = σ(x) σ(y) - µ(x,y) x,y S
Using (17), µc(x,y) = σ' [h(x)] σ' [h(y)] (27)Using (24) and (25) in (27) we get,
–µ'[h(x),h(y)] x,y S
Using (18), µc(x,y) = µc' [h(x),h(y)] x,y S

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µc[h-1(x'), h-1 (y')]≥ σ' [h(x)] σ' [h(y)] - µ' Remark:5.5 As a consequence of the above theorm
[h(x),h(y)] x',y' S' ≥ σ' (x') σ' (y') - µ' if G is a self complementary fuzzy graph, then
(x',y')
Size of = q=
c
≥ µ '(x',y') x',y' S'[by using
(27)] This theorem is not sufficient. Even though G is
not isomorphic to G', G satisfies the equality
That is, µc' (x',y') ≤ µc[h-1(x'), h-1 (y')] (x,y) = x,y S.
(28)

Thus h-1 : S'→S is a bijective map, which is a weak
VI.SELF WEAK COMPLEMENTARY
isomorphismbetweenGcandGc'by(26)and(28). FUZZY GRAPHS
Definition:6.1
V.SELF COMPLEMENTARY FUZZY
GRAPHS A fuzzy graph G is self weak complementary if G
Definition: 5.1 is weak isomorphic with Gc. Where Gc is the
A fuzzy graph G is self complementary if G ≅G'. complement of fuzzy graph G.
Where G' is the complement of fuzzy grph G.
σ(x) = σ[h(x)], x S
Example:5.2 (referFig: 5.1)
µ(x, y) ≤ µc[h(x), h(y)], x, y S
Example: 5.3(referFig: 5.2)
Remark:6.2 The fuzzy graph G, fig5.2 give in
Theorem:5.4
example 5.3 is not a self weak complementary
G : (σ, µ) be a self complementary fuzzy graph, fuzzy graph.
then
Example:6.3(refer Fig. 6.1)
(x,y) = x,y S.
Theorem:6.4
Proof:
G : (σ, µ) be a self weak complementary fuzzy
Let G : (σ, µ) be a self complementary fuzzy graph. graph, then (x,y) ≤
Then there exists an isomorphism h : S→S
such that, h(x) = x x S satisfying, x,y S

σc[h(x)] = σ[h(x)] = σ(x) x S Proof: Let G : (σ, µ) be a self weak
complementary fuzzy graph. Then there exists a
(29)
weak isomorphism h : S→S such that, h(x) =
µc[h(x), h(y)] = µ[h(x), h(y)] = µ(x,y) x,y S x satisfying,
(30)
σ (x) = σ[h(x)] x S
we have, (31)
µc[h(x), h(y)] = σc [h(x)] σc[h(y)] - µ[h(x), h(y)]
µ(x, y) ≤ µc[h(x), h(y)] x, y S
ie., µ(x,y) = σ(x) σ(y) - µ[h(x), h(y)]
(32)
(29)+( 30)⟹
(x,y) + Now by definition(2.3), we have,
[h(x),h(y)]= µc[h(x), h(y)] = σ[h(x)] σ[h(y)] - µ[h(x), h(y)]
Using equation (2) (33)
2 (x,y) = from (32) and (33),
(x,y) = . x,y S µ(x,y) ≤ σ[h(x)] σ[h(y)] - µ[h(x), h(y)]
using (31) in this above equation,
µ(x,y) ≤ σ(x) σ(y) - µ[h(x), h(y)]
Hence G : (σ, µ) be a self complementary fuzzy
graph. Then µ(x,y) + µ[h(x), h(y)] ≤ σ(x) σ(y)
Taking summation on both sides, we have,
(x,y)= x,y S. (x,y) + [h(x), h(y)] ≤ (x)
σ(y)] (Since S is a finite set)

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Therefore we have, b(1) 0.5 c(1) a’(1) 0.25
2 [h(x),h(y)]≤ (x) σ(y)][since h(x) = b’(1)
x]
G : (σ,µ) G’ : (σ’,µ’)
Hence [h(x), h(y)] ≤ (x) σ(y)]in
S. Figure:3.1 Graphs of same order and size
Theorem:6.5
Fig.4.1
If µ(x,y) ≤ x,y S then G is self
weak complementary.WhereG:(σ,µ) is a fuzzy
graph. a (1) 0.25 b(0.75) d’(0.25) 0.5
Proof: c’(1)

Given G : (σ,µ) is a fuzzy graph and 0.5 00 0.75 0.75
0.75
µ(x, y) ≤
d(0.25) 0.125 c(1) a’(1) 0.25 b’(0.75
(34)
G : (σ,µ) G’ : (σ’,µ’)
Consider the identity map h : S→S such that
h(x) = x, x S and Fig:4.1 Graphs of same order and size but not
weak isomorphic
σ (x) = σ[h(x)] x S
(35) The above two fuzzy graphs are of order 3 but
they are not weak isomorphic.
Now by definition(2.3), we have,
µc(x, y) = σ(x) σ(y) - µ(x, y) Fig.4.2
(36)
Using (34) in (36), c(1/3) c'(1/3)
µc(x,y)≥σ(x) σ(y)- [ by equation
(34)] 1/4 1/5 1/3 1/4
µc(x, y) ≥
µc(x, y) ≥ µ(x, y) x S[ by equation (35)]
That is, µ(x, y)≤µ (x,y)=µc[h(x),h(y)][ since h(x) =
c

x]
Therefore, µ(x, y) ≤ µc[h(x), h(y)] x S a(1/2) 1/5 b(1/4) a' (1/2) 1/4 b'
(1/4)
(37)
From equation (34) and (35) G is weak isomorphic G :(σ,µ) G' :(σ',µ')
with Gc. Therefore G is a self weak complementary
fuzzy graph. Figure: 4.2 Weak isomorphism
Remark:6.6 Fig.5.1
When G is a self weak complementary graph then a(0.6) a(0.6)
order(G) =order (Gc) and size(G) ≤ size(Gc)
But the converse of the above is not true. 0.3 0.3 0.3 0.3
As in example , we have order(G) =order (Gc) = 2.8
and size(G)= 1.6 and size(Gc) = 1.6. but G is not a
self weak complementary fuzzy graph.
VII. FIGURES
Fig.3.1

a(1) d(1) d’(1) 0.25
c’(1) b(0.8) 0.4 c(1) b(0.8) 0.4 c(1)

0.25 0.25 G :(σ,µ) Gc :(σ,µc)
0.5 Fig: 5.1self complementary fuzzy graph

In this example G ≅ Gc.

1632 | P a g e

Fig.5.2 [5] Mordeson, J.N. and P.S. Nair Fuzzy
c(0.6) 0.3 b(1) c(0.6) 0.3 Graphs and Fuzzy Hypergraphs Physica
b(1) Verlag, Heidelberg 1998; Second Edition
2001.
03 0.6 0.1 0.4 0.6 [6] Nagoorgani, A., and Chandrasekaran,
0.2 V.T., Domination in fuzzy graph,Adv. in
0 Fuzzy sets & Systems 1(1) (2006), 17-26.
[7] Nagoorgani. A., and Basheer Ahamed, M.,
Strong and WeakDomination in Fuzzy
Graphs, East Asian Mathematical Journal,
d(0.4) 0.4 a(0.8) d(0.4) Vol.23,No.1, June 30, (2007) pp 1-8.
a(0.8)

(i) G :(σ,µ) (ii) Gc (σ,µc)
Fig:5.2 Fuzzy graphs are not Self complement

In this example G ≇ Gc.

Fig.6.1

d(0.4) 0.3 c(0.6) d (0.4) 0.3 c(0.6)

0.2 0.1 0.2 0.2 0.4 0.5 0.4

a(0.8) 0.3 b(1) a (0.8) 0.5 b(1)

(i) G :(σ,µ) (ii) Gc :(σ,µc

Fig :6.1 self weak complementary fuzzy graph.

Conclusion
In this paper isomorphism between fuzzy graphs
is proved to be an equivalence relation and weak
isomorphism is proved. To be a partial order
relation. A necessary and then a sufficient
condition for a fuzzy graph to be self weak
complementary are studied. The results discussed
may be used to study about various fuzzy graph
invariants.

REFERENCES
1] Bhattacharya, P, Some Remarks on fuzzy
graphs, Pattern Recognition Letter 6: 297-
302, 1987.
[2] Bhutani, K.R., On Automorphism of Fuzzy
graphs, Pattern Recognition Letter 9: 159-
162,1989
[3] Sunitha, M.S., and Vijayakumar,.A,
Complement of fuzzy graph,Indian J,.pure
appl, Math.,33(9); 1451-1464 September
2002.
[4] Somasundaram. A., and Somasundaram, S.,
Domination in fuzzygraphs, Pattern
Recognition Letter 19: (1998) 787-791.

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