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=
(
,
)
Now, for the graph
G
V
H
we can construct the Extended Intuitionistic
Fuzzy Graph (EIFG)
G
∗
=
(
V
∗
,
H
∗
)
. It has the following IM-representation:
V
∗
,
V
∗
,
{
μ(v
i
,v
j
), ν(v
i
,v
j
)
}]
[
v
1
,
α(v
1
), β(v
1
)
... v
n
,
α(v
n
), β(v
n
)
v
1
,
α(v
1
), β(v
1
)
μ
v
1
,v
1
,ν
v
1
,v
1
...
μ
v
1
,v
n
,ν
v
1
,v
n
.
.
.
.
...
=
,
v
i
,
α(v
i
), β(v
i
)
μ
v
i
,v
1
,ν
v
i
,v
1
...
μ
v
i
,v
n
,ν
v
i
,v
n
.
.
.
.
...
v
n
,
α(v
n
), β(v
n
)
μ
v
n
,v
1
,ν
v
n
,v
1
...
μ
v
n
,v
n
,ν
v
n
,v
n
where for every 1
≤
i
≤
n
,
1
≤
j
≤
n
:
μ
v
i
,v
j
,ν
v
i
,v
j
∈[
0
,
1
]
,μ
v
i
,v
j
+
ν
v
i
,v
j
∈[
0
,
1
]
,
α(v
i
), β(v
i
)
∈[
]
.
Let us discuss here for simplicity only the case of oriented graph. Let us denote
by
x
0
,
1
]
,α(v
i
)
+
β(v
i
)
∈[
0
,
1
y
the fact that both vertices
x
and
y
are connected by an arc and
x
is higher
than
y
. Let operation
→
,
×}
.
We call that the EIFG
G
∗
is “well-top-down-(very strong, strong, middle, weak,
veryweak)-ordered”, or shortly, “well-top-down-
◦∈{+
,
max
,
@
,
min
◦
-ordered”, if for every two vertices
v
i
and
v
j
, such that
v
i
→
v
j
, the following inequality holds:
α
i
,β
i
◦
μ
v
i
,v
j
,ν
v
i
,v
j
≥
α
j
,β
j
.
Analogously, we call that the EIFG
G
∗
is “well-bottom-up-(very strong, strong,
middle, weak, very weak)-ordered”, or shortly, “well-bottom-up-
◦
-ordered”, if for
every two vertices
v
i
and
v
i
→
v
j
, the following inequality holds:
α
i
,β
i
◦
μ
v
i
,v
j
,ν
v
i
,v
j
≤
α
j
,β
j
.
v
j
, such that
We illustrate the way for IM-interpretation of the EIFGs by the following example.
Let us have the EIFG
G
∗
with the form
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