<s>
In	O
combinatorial	O
optimization	O
,	O
the	O
Gomory	B-Algorithm
–	I-Algorithm
Hu	I-Algorithm
tree	I-Algorithm
of	O
an	O
undirected	O
graph	O
with	O
capacities	O
is	O
a	O
weighted	O
tree	O
that	O
represents	O
the	O
minimum	O
s-t	B-Algorithm
cuts	I-Algorithm
for	O
all	O
s-t	O
pairs	O
in	O
the	O
graph	O
.	O
</s>
<s>
The	O
Gomory	B-Algorithm
–	I-Algorithm
Hu	I-Algorithm
tree	I-Algorithm
can	O
be	O
constructed	O
in	O
maximum	B-Algorithm
flow	I-Algorithm
computations	O
.	O
</s>
<s>
Denote	O
the	O
minimum	O
capacity	O
of	O
an	O
s-t	B-Algorithm
cut	I-Algorithm
by	O
λst	O
for	O
each	O
s	O
,	O
t	O
∈	O
VG	O
.	O
</s>
<s>
Se	O
,	O
Te	O
⊆	O
VG	O
are	O
the	O
two	O
connected	O
components	O
of	O
T∖{e},	O
and	O
thus	O
(	O
Se	O
,	O
Te	O
)	O
form	O
an	O
s-t	B-Algorithm
cut	I-Algorithm
in	O
G	O
.	O
</s>
<s>
Output	O
:	O
A	O
Gomory	B-Algorithm
–	I-Algorithm
Hu	I-Algorithm
Tree	I-Algorithm
T	O
=	O
(	O
VT	O
,	O
ET	O
)	O
.	O
</s>
<s>
Choose	O
two	O
vertices	O
s	O
,	O
t	O
∈	O
X	O
and	O
find	O
a	O
minimum	O
s-t	B-Algorithm
cut	I-Algorithm
(	O
A	O
 '	O
,	O
B	O
 '	O
)	O
in	O
G	O
 '	O
.	O
</s>
<s>
Then	O
it	O
can	O
be	O
shown	O
that	O
the	O
minimum	O
s-t	B-Algorithm
cut	I-Algorithm
in	O
G	O
 '	O
is	O
also	O
a	O
minimum	O
s-t	B-Algorithm
cut	I-Algorithm
in	O
G	O
for	O
any	O
s	O
,	O
t	O
∈	O
X	O
.	O
</s>
<s>
The	O
Lemma	O
can	O
be	O
used	O
again	O
repeatedly	O
to	O
show	O
that	O
the	O
output	O
T	O
satisfies	O
the	O
properties	O
of	O
a	O
Gomory	B-Algorithm
–	I-Algorithm
Hu	I-Algorithm
Tree	I-Algorithm
.	O
</s>
<s>
red	O
and	O
blue	O
coloring	O
represents	O
the	O
s-t	B-Algorithm
cut	I-Algorithm
.	O
</s>
<s>
The	O
minimum	O
s-t	B-Algorithm
cut	I-Algorithm
(	O
A	O
 '	O
,	O
B	O
 '	O
)	O
is	O
( {	O
0	O
,	O
1	O
,	O
2	O
,	O
4}	O
,	O
{	O
3	O
,	O
5}	O
)	O
with	O
c'( A'	O
,	O
B	O
 '	O
)	O
=	O
6	O
.	O
</s>
<s>
The	O
minimum	O
s-t	B-Algorithm
cut	I-Algorithm
(	O
A	O
 '	O
,	O
B	O
 '	O
)	O
in	O
G	O
 '	O
is	O
(	O
{{	O
0	O
,	O
1	O
,	O
2	O
,	O
4}	O
,	O
3}	O
,	O
 { 5 } 	O
)	O
with	O
c'( A'	O
,	O
B	O
 '	O
)	O
=	O
8	O
.	O
</s>
<s>
The	O
minimum	O
s-t	B-Algorithm
cut	I-Algorithm
(	O
A	O
 '	O
,	O
B	O
 '	O
)	O
in	O
G	O
 '	O
is	O
(	O
{	O
1	O
,	O
{	O
3	O
,	O
5}	O
,	O
4}	O
,	O
{	O
0	O
,	O
2}	O
)	O
with	O
c'( A'	O
,	O
B	O
 '	O
)	O
=	O
6	O
.	O
</s>
<s>
The	O
minimum	O
s-t	B-Algorithm
cut	I-Algorithm
(	O
A	O
 '	O
,	O
B	O
 '	O
)	O
in	O
G	O
 '	O
is	O
(	O
{	O
1	O
,	O
{	O
3	O
,	O
5}}	O
,	O
{{	O
0	O
,	O
2}	O
,	O
4}	O
)	O
with	O
c'( A'	O
,	O
B	O
 '	O
)	O
=	O
7	O
.	O
</s>
<s>
The	O
minimum	O
s-t	B-Algorithm
cut	I-Algorithm
(	O
A	O
 '	O
,	O
B	O
 '	O
)	O
in	O
G	O
 '	O
is	O
(	O
{0},	O
{	O
2	O
,	O
{	O
1	O
,	O
3	O
,	O
4	O
,	O
5}}	O
)	O
with	O
c'( A'	O
,	O
B	O
 '	O
)	O
=	O
8	O
.	O
</s>
<s>
Gusfield	O
's	O
algorithm	O
can	O
be	O
used	O
to	O
find	O
a	O
Gomory	B-Algorithm
–	I-Algorithm
Hu	I-Algorithm
tree	I-Algorithm
without	O
any	O
vertex	O
contraction	O
in	O
the	O
same	O
running	O
time-complexity	O
,	O
which	O
simplifies	O
the	O
implementation	O
of	O
constructing	O
a	O
Gomory	B-Algorithm
–	I-Algorithm
Hu	I-Algorithm
Tree	I-Algorithm
.	O
</s>
<s>
Andrew	O
V	O
.	O
Goldberg	O
and	O
K	O
.	O
Tsioutsiouliklis	O
implemented	O
the	O
Gomory-Hu	B-Algorithm
algorithm	I-Algorithm
and	O
Gusfield	O
algorithm	O
,	O
and	O
performed	O
an	O
experimental	O
evaluation	O
and	O
comparison	O
.	O
</s>
<s>
In	O
planar	O
graphs	O
,	O
the	O
Gomory	B-Algorithm
–	I-Algorithm
Hu	I-Algorithm
tree	I-Algorithm
is	O
dual	O
to	O
the	O
minimum	O
weight	O
cycle	O
basis	O
,	O
in	O
the	O
sense	O
that	O
the	O
cuts	O
of	O
the	O
Gomory	B-Algorithm
–	I-Algorithm
Hu	I-Algorithm
tree	I-Algorithm
are	O
dual	O
to	O
a	O
collection	O
of	O
cycles	O
in	O
the	O
dual	O
graph	O
that	O
form	O
a	O
minimum-weight	O
cycle	O
basis	O
.	O
</s>
