Assignment 10
Solutions
1. (10 points)
Consider the following bipartite graph
G
with bipartition (
A,B
)
where
A
=
{
1
,...,
9
}
and
B
=
{
a,b,.
..,i
}
.
i
2
3
4
5
6
7
8
9
a
b
c
d
e
f
g
h
1
i. Let
M
=
{
3
a,
4
b,
6
c,
7
d,
9
h,
8
i
}
. Using the bipartite matching algorithm, de
termine the sets
X
0
,
X
, and
Y
from the XY construction. Find either an
M
augmenting path or cover
C
with

C

=

M

.
Solution.
X
0
=
{
1
,
2
,
5
}
. So 1
,
2
,
5
∈
X
. The following table gives the vertices
in
X
and
Y
in the order added.
new vertex
a
3
c
6
b
4
d
7
e
h
9
g
f
i
8
added to set
Y X Y X Y X Y X Y Y X Y Y Y X
predecessor
1
a
5
c
3
b
5
d
7
7
h
7
6
9
i
Thus
X
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
and
Y
=
{
a,b,c,d,e,f,g,h,i
}
. Note that
g
∈
Y
is
M
exposed, and
P
= (5
,d,
7
,g
) is an
M
augmenting path.
ii. Let
M
=
{
1
a,
2
c,
5
d,
4
b,
6
f,
8
i,
9
h,
7
e
}
. Using the bipartite matching algorithm,
determine the sets
X
0
,
X
, and
Y
from the XY construction. Find either an
M
augmenting path or cover
C
with

C

=

M

.
Solution.
X
0
=
{
3
}
. So 3
∈
X
. The following table gives the vertices in
X
and
Y
in the order added.
new vertex
a
b
c
d
1
2
4
5
added to set
Y Y Y Y X X X X
predecessor
3
3
3
3
a
c
b
d
Thus
X
=
{
1
,
2
,
3
,
4
,
5
}
and
Y
=
{
a,b,c,d
}
. Note that
Y
contains no
M

exposed vertex. Let
C
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 Spring '09
 M.PEI
 Math, Graph Theory, Bipartite graph, Dulmage–Mendelsohn decomposition, v0, nonnegative integer entries

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