<s>
In	O
graph	O
theory	O
,	O
the	O
Dulmage	B-Algorithm
–	I-Algorithm
Mendelsohn	I-Algorithm
decomposition	I-Algorithm
is	O
a	O
partition	O
of	O
the	O
vertices	O
of	O
a	O
bipartite	O
graph	O
into	O
subsets	O
,	O
with	O
the	O
property	O
that	O
two	O
adjacent	O
vertices	O
belong	O
to	O
the	O
same	O
subset	O
if	O
and	O
only	O
if	O
they	O
are	O
paired	O
with	O
each	O
other	O
in	O
a	O
perfect	O
matching	O
of	O
the	O
graph	O
.	O
</s>
<s>
A	O
generalization	O
to	O
any	O
graph	O
is	O
the	O
Edmonds	B-Algorithm
–	I-Algorithm
Gallai	I-Algorithm
decomposition	I-Algorithm
,	O
using	O
the	O
Blossom	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
For	O
each	O
component	O
of	O
H	O
,	O
form	O
a	O
subset	O
of	O
the	O
Dulmage	B-Algorithm
–	I-Algorithm
Mendelsohn	I-Algorithm
decomposition	I-Algorithm
consisting	O
of	O
the	O
vertices	O
in	O
G	O
that	O
are	O
endpoints	O
of	O
edges	O
in	O
the	O
component	O
.	O
</s>
<s>
As	O
another	O
component	O
of	O
the	O
Dulmage	B-Algorithm
–	I-Algorithm
Mendelsohn	I-Algorithm
decomposition	I-Algorithm
,	O
Dulmage	O
and	O
Mendelsohn	O
defined	O
the	O
core	O
of	O
a	O
graph	O
to	O
be	O
the	O
union	O
of	O
its	O
maximum	O
matchings	O
.	O
</s>
<s>
However	O
,	O
this	O
concept	O
should	O
be	O
distinguished	O
from	O
the	O
core	O
in	O
the	O
sense	O
of	O
graph	O
homomorphisms	O
,	O
and	O
from	O
the	O
k-core	B-Algorithm
formed	O
by	O
the	O
removal	O
of	O
low-degree	O
vertices	O
.	O
</s>
<s>
This	O
decomposition	O
has	O
been	O
used	O
to	O
partition	O
meshes	O
in	O
finite	B-Application
element	I-Application
analysis	I-Application
,	O
and	O
to	O
determine	O
specified	O
,	O
underspecified	O
and	O
overspecified	O
equations	O
in	O
systems	O
of	O
nonlinear	O
equations	O
.	O
</s>
