<s>
Here	O
a	O
surface	O
is	O
a	O
compact	O
,	O
connected	O
-manifold	O
.	O
</s>
<s>
Some	O
authors	O
define	O
a	O
weaker	O
version	O
of	O
the	O
definition	O
of	O
"	O
graph	B-Algorithm
embedding	I-Algorithm
"	O
by	O
omitting	O
the	O
non-intersection	O
condition	O
for	O
edges	O
.	O
</s>
<s>
In	O
such	O
contexts	O
the	O
stricter	O
definition	O
is	O
described	O
as	O
"	O
non-crossing	O
graph	B-Algorithm
embedding	I-Algorithm
"	O
.	O
</s>
<s>
This	O
article	O
deals	O
only	O
with	O
the	O
strict	O
definition	O
of	O
graph	B-Algorithm
embedding	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
the	O
graph	B-Algorithm
genus	I-Algorithm
is	O
NP-hard	O
(	O
the	O
problem	O
of	O
determining	O
whether	O
an	O
-vertex	O
graph	O
has	O
genus	O
is	O
NP-complete	O
)	O
.	O
</s>
<s>
At	O
the	O
same	O
time	O
,	O
the	O
graph	B-Algorithm
genus	I-Algorithm
problem	O
is	O
fixed-parameter	O
tractable	O
,	O
i.e.	O
,	O
polynomial	O
time	O
algorithms	O
are	O
known	O
to	O
check	O
whether	O
a	O
graph	O
can	O
be	O
embedded	O
into	O
a	O
surface	O
of	O
a	O
given	O
fixed	O
genus	O
as	O
well	O
as	O
to	O
find	O
the	O
embedding	O
.	O
</s>
