Biology Reference
In-Depth Information
ˆ
N
(
w
1
)
(
w
1
)
Let
|
∪
V
|
=2
p
for the natural number
p
and let
k
be such that
2
k
.Define
2
k
>p
. Label
k
lexicographically as
h
1
,
h
2
,...,
h
{−
1
,
1
}
η
:
¯
n
+
k
so that
V→{−
1
,
1
}
(
η
(
v
)
,
1
,
1
,...,
1)
,
∈V
\
T
(
w
1
)
v
(
η
(
v
i
)
,
h
i
+1
)
,
v
=
v
i
,
1
T
(
w
1
)
≤
i
≤|
|
η
(
v
i−|T
(
w
1
)
|
)
,
h
2
k
T
(
w
1
)
−
i
+
|
|
)
,v
=
v
i
,
|
(
−
T
(
w
1
)
T
(
w
1
)
|
+1
≤
i
≤
2
|
|
i−|T
(
w
1
)
|
)
,
(
η
(
v
i
)
,
h
v
=
v
i
,
2
v
→
ˆ
T
(
w
1
)
(
w
1
)
|
|
+1
≤
i
≤|
V
|
,
i
odd
2
k
−i
+2
|T
(
w
1
)
|
)
,
η
(
v
i−
1
)
,
h
v
=
v
i
,
2
(
−
ˆ
T
(
w
1
)
(
w
1
)
|
|
+1
≤
i
≤|
V
|
,
i
even
n
+
k
so that
and
γ
:
W→{−
2
,
0
,
2
}
(
γ
(
w
)
,
2
,
2
,...,
2)
,w
∈W
w
→
(0
,
0
,...,
0)
,
w
=
w
1
.
We conclude this section by showing that any bipartite graph, including those
with isolated nodes, can be extended and labeled to become a diversity graph. The
result is an extension of Lemma 6.2, but the edges added between isolated nodes
are handled outside the definition of
¯
E
.
Theorem 6.5.
Any bipartite graph
(
)
can be extended and labeled to be-
come a diversity graph by adding no more than
V
,
W
,
E
ˆ
|
V
(
w
)
|
+(2
M
W
−
M
V
)
+
+
M
V
(mod2)
w∈W
nodes to
V
,where
M
V
and
M
W
are the number of isolated nodes in
V
and
W
,
respectively.
V
,
W
,
E
) be
Proof.
Let
V
I
and
W
I
be the isolated nodes in
V
and
W
and let (
V
,
W
,
E
) as in
the subgraph of (
V
,
W
,
E
) with these nodes removed. Extend (
Lemma6.2 so that (
¯
W
,
¯
V
,
E
,η
,γ
) is a diversity graph. Since
N
(
w
)=
T
(
w
)=
∅
for all
w
∈W
I
,wehavethat
w∈W
|
=
w∈W
ˆ
ˆ
V
(
w
)
|
|
V
(
w
)
|
,
E
) to (
¯
W
,
¯
V
,
W
,
V
,
E
) adds
and
we
conclude
that
the
extension
of (
w∈W
|
ˆ
).Let
n
be such that the images of
η
and
γ
V
(
w
)
|
nodes to (
V
,
W
,
E
n
and
n
.Let
k
be a natural number so that 2
k
>
are in
{−
1
,
1
}
{−
2
,
0
,
2
}
|W
I
|
.
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