<s>
In	O
graph	O
theory	O
,	O
the	O
Gallai	B-Algorithm
–	I-Algorithm
Edmonds	I-Algorithm
decomposition	I-Algorithm
is	O
a	O
partition	O
of	O
the	O
vertices	O
of	O
a	O
graph	O
into	O
three	O
subsets	O
which	O
provides	O
information	O
on	O
the	O
structure	O
of	O
maximum	O
matchings	O
in	O
the	O
graph	O
.	O
</s>
<s>
The	O
Gallai	B-Algorithm
–	I-Algorithm
Edmonds	I-Algorithm
decomposition	I-Algorithm
of	O
a	O
graph	O
can	O
be	O
found	O
using	O
the	O
blossom	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Given	O
a	O
graph	O
,	O
its	O
Gallai	B-Algorithm
–	I-Algorithm
Edmonds	I-Algorithm
decomposition	I-Algorithm
consists	O
of	O
three	O
disjoint	B-Algorithm
sets	I-Algorithm
of	O
vertices	O
,	O
,	O
,	O
and	O
,	O
whose	O
union	O
is	O
:	O
the	O
set	O
of	O
all	O
vertices	O
of	O
.	O
</s>
<s>
The	O
Gallai	B-Algorithm
–	I-Algorithm
Edmonds	I-Algorithm
decomposition	I-Algorithm
has	O
the	O
following	O
properties	O
:	O
</s>
<s>
The	O
Gallai	B-Algorithm
–	I-Algorithm
Edmonds	I-Algorithm
decomposition	I-Algorithm
is	O
a	O
generalization	O
of	O
Dulmage	B-Algorithm
–	I-Algorithm
Mendelsohn	I-Algorithm
decomposition	I-Algorithm
from	O
bipartite	O
graphs	O
to	O
general	O
graphs	O
.	O
</s>
<s>
An	O
extension	O
of	O
the	O
Gallai	B-Algorithm
–	I-Algorithm
Edmonds	I-Algorithm
decomposition	I-Algorithm
theorem	O
to	O
multi-edge	O
matchings	O
is	O
given	O
in	O
Katarzyna	O
Paluch	O
's	O
"	O
Capacitated	O
Rank-Maximal	O
Matchings	O
"	O
.	O
</s>
