<s>
A	O
combinatorial	B-Data_Structure
map	I-Data_Structure
is	O
a	O
combinatorial	O
representation	O
of	O
a	O
graph	O
on	O
an	O
orientable	O
surface	O
.	O
</s>
<s>
A	O
combinatorial	B-Data_Structure
map	I-Data_Structure
may	O
also	O
be	O
called	O
a	O
combinatorial	O
embedding	O
,	O
a	O
rotation	O
system	O
,	O
an	O
orientable	O
ribbon	O
graph	O
,	O
a	O
fat	O
graph	O
,	O
or	O
a	O
cyclic	O
graph	O
.	O
</s>
<s>
Combinatorial	B-Data_Structure
maps	I-Data_Structure
are	O
used	O
as	O
efficient	O
data	O
structures	O
in	O
image	O
representation	O
and	O
processing	B-Algorithm
,	O
in	O
geometrical	O
modeling	O
.	O
</s>
<s>
A	O
combinatorial	B-Data_Structure
map	I-Data_Structure
is	O
a	O
boundary	B-Algorithm
representation	I-Algorithm
model	O
;	O
it	O
represents	O
object	O
by	O
its	O
boundaries	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
combinatorial	B-Data_Structure
map	I-Data_Structure
was	O
introduced	O
informally	O
by	O
J	O
.	O
Edmonds	O
for	O
polyhedral	O
surfaces	O
which	O
are	O
planar	O
graphs	O
.	O
</s>
<s>
The	O
term	O
"	O
constellation	O
"	O
was	O
not	O
retained	O
and	O
instead	O
"	O
combinatorial	B-Data_Structure
map	I-Data_Structure
"	O
was	O
favored	O
.	O
</s>
<s>
Combinatorial	B-Data_Structure
maps	I-Data_Structure
were	O
later	O
generalized	O
to	O
represent	O
higher-dimensional	O
orientable	O
subdivided	O
objects	O
.	O
</s>
<s>
When	O
all	O
the	O
represented	O
cells	O
are	O
simplexes	O
,	O
a	O
simplicial	O
complex	O
may	O
be	O
used	O
,	O
but	O
when	O
we	O
want	O
to	O
represent	O
any	O
type	O
of	O
cells	O
,	O
we	O
need	O
to	O
use	O
cellular	O
topological	O
models	O
like	O
combinatorial	B-Data_Structure
maps	I-Data_Structure
or	O
generalized	O
maps	O
.	O
</s>
<s>
A	O
combinatorial	B-Data_Structure
map	I-Data_Structure
is	O
a	O
triplet	O
such	O
that	O
:	O
</s>
<s>
is	O
an	O
involution	B-Algorithm
on	O
with	O
no	O
fixed	O
point	O
.	O
</s>
<s>
Intuitively	O
,	O
a	O
combinatorial	B-Data_Structure
map	I-Data_Structure
corresponds	O
to	O
a	O
graph	O
where	O
each	O
edge	O
is	O
subdivided	O
into	O
two	O
darts	O
(	O
sometimes	O
also	O
called	O
half-edges	O
)	O
.	O
</s>
<s>
So	O
,	O
there	O
are	O
two	O
ways	O
to	O
represent	O
a	O
combinatorial	B-Data_Structure
map	I-Data_Structure
depending	O
if	O
the	O
permutation	O
is	O
or	O
(	O
see	O
example	O
below	O
)	O
.	O
</s>
<s>
+Combinatorial	O
maps	O
example	O
:	O
a	O
plane	O
graph	O
and	O
the	O
two	O
combinatorial	B-Data_Structure
maps	I-Data_Structure
depending	O
if	O
we	O
use	O
the	O
notation	O
or	O
.thumb	O
|center|200px|A	O
plane	O
graphthumb|center|200px|Corresponding	O
combinatorial	B-Data_Structure
map	I-Data_Structure
.	O
</s>
<s>
Darts	O
are	O
represented	O
by	O
numbered	O
segments	O
,	O
by	O
gray	O
arrows	O
(	O
example	O
)	O
,	O
two	O
darts	O
linked	O
by	O
are	O
drawn	O
consecutively	O
and	O
separated	O
by	O
a	O
small	O
bar	O
(	O
example	O
)	O
.thumb	O
|center|200px|Corresponding	O
combinatorial	B-Data_Structure
map	I-Data_Structure
.	O
</s>
<s>
An	O
n-dimensional	O
combinatorial	B-Data_Structure
map	I-Data_Structure
(	O
or	O
n-map	B-Data_Structure
)	O
is	O
a	O
(	O
n+1	O
)	O
-tuple	O
such	O
that	O
:	O
</s>
<s>
are	O
involutions	B-Algorithm
on	O
;	O
</s>
<s>
is	O
an	O
involution	B-Algorithm
if	O
.	O
</s>
<s>
An	O
n-dimensional	O
combinatorial	B-Data_Structure
map	I-Data_Structure
represents	O
the	O
subdivision	O
of	O
a	O
closed	O
orientable	O
n-dimensional	O
space	O
.	O
</s>
<s>
Two-dimensional	O
combinatorial	B-Data_Structure
maps	I-Data_Structure
can	O
be	O
retrieved	O
by	O
fixing	O
and	O
renaming	O
by	O
and	O
by	O
.	O
</s>
