<s>
In	O
constraint	B-Application
satisfaction	I-Application
research	O
in	O
artificial	B-Application
intelligence	I-Application
and	O
operations	O
research	O
,	O
constraint	B-Application
graphs	I-Application
and	O
hypergraphs	O
are	O
used	O
to	O
represent	O
relations	O
among	O
constraints	O
in	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
.	O
</s>
<s>
A	O
constraint	B-Application
graph	I-Application
is	O
a	O
special	O
case	O
of	O
a	O
factor	O
graph	O
,	O
which	O
allows	O
for	O
the	O
existence	O
of	O
free	O
variables	O
.	O
</s>
<s>
The	O
constraint	B-Application
hypergraph	I-Application
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
a	O
hypergraph	O
in	O
which	O
the	O
vertices	O
correspond	O
to	O
the	O
variables	O
,	O
and	O
the	O
hyperedges	O
correspond	O
to	O
the	O
constraints	O
.	O
</s>
<s>
A	O
simple	O
way	O
to	O
represent	O
the	O
constraint	B-Application
hypergraph	I-Application
is	O
by	O
using	O
a	O
classical	O
graph	O
with	O
the	O
following	O
properties	O
:	O
</s>
<s>
The	O
primal	B-Application
constraint	I-Application
graph	I-Application
or	O
simply	O
primal	O
graph	O
(	O
also	O
the	O
Gaifman	O
graph	O
)	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
the	O
graph	O
whose	O
nodes	O
are	O
the	O
variables	O
of	O
the	O
problem	O
and	O
an	O
edge	O
joins	O
a	O
pair	O
of	O
variables	O
if	O
the	O
two	O
variables	O
occur	O
together	O
in	O
a	O
constraint	O
.	O
</s>
<s>
The	O
primal	O
constraint	O
graph	O
is	O
in	O
fact	O
the	O
primal	O
graph	O
of	O
the	O
constraint	B-Application
hypergraph	I-Application
.	O
</s>
<s>
The	O
dual	B-Application
constraint	I-Application
graph	I-Application
is	O
the	O
graph	O
in	O
which	O
the	O
vertices	O
are	O
all	O
constraint	O
scopes	O
involved	O
in	O
the	O
constraints	O
of	O
the	O
problem	O
,	O
and	O
two	O
vertices	O
are	O
connected	O
by	O
an	O
edge	O
if	O
the	O
corresponding	O
scopes	O
have	O
common	O
variables	O
.	O
</s>
