<s>
Bidimensionality	B-General_Concept
theory	O
characterizes	O
a	O
broad	O
range	O
of	O
graph	O
problems	O
(	O
bidimensional	O
)	O
that	O
admit	O
efficient	O
approximate	O
,	O
fixed-parameter	O
or	O
kernel	O
solutions	O
in	O
a	O
broad	O
range	O
of	O
graphs	O
.	O
</s>
<s>
These	O
graph	O
classes	O
include	O
planar	O
graphs	O
,	O
map	O
graphs	O
,	O
bounded-genus	B-Algorithm
graphs	O
and	O
graphs	O
excluding	O
any	O
fixed	O
minor	O
.	O
</s>
<s>
In	O
particular	O
,	O
bidimensionality	B-General_Concept
theory	O
builds	O
on	O
the	O
graph	O
minor	O
theory	O
of	O
Robertson	O
and	O
Seymour	O
by	O
extending	O
the	O
mathematical	O
results	O
and	O
building	O
new	O
algorithmic	O
tools	O
.	O
</s>
<s>
A	O
parameterized	B-General_Concept
problem	I-General_Concept
is	O
a	O
subset	O
of	O
for	O
some	O
finite	O
alphabet	O
.	O
</s>
<s>
An	O
instance	O
of	O
a	O
parameterized	B-General_Concept
problem	I-General_Concept
consists	O
of	O
(	O
x	O
,	O
k	O
)	O
,	O
where	O
k	O
is	O
called	O
the	O
parameter	O
.	O
</s>
<s>
Examples	O
of	O
minor-bidimensional	O
problems	O
are	O
the	O
parameterized	B-General_Concept
versions	O
of	O
vertex	O
cover	O
,	O
feedback	O
vertex	O
set	O
,	O
minimum	O
maximal	O
matching	O
,	O
and	O
longest	O
path	O
.	O
</s>
<s>
All	O
algorithmic	O
applications	O
of	O
bidimensionality	B-General_Concept
are	O
based	O
on	O
the	O
following	O
combinatorial	O
property	O
:	O
either	O
the	O
treewidth	O
of	O
a	O
graph	O
is	O
small	O
,	O
or	O
the	O
graph	O
contains	O
a	O
large	O
grid	O
as	O
a	O
minor	O
or	O
contraction	O
.	O
</s>
<s>
Bidimensionality	B-General_Concept
theory	O
has	O
been	O
used	O
to	O
obtain	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
schemes	I-Algorithm
for	O
many	O
bidimensional	O
problems	O
.	O
</s>
<s>
If	O
a	O
minor	O
(	O
contraction	O
)	O
bidimensional	O
problem	O
has	O
several	O
additional	O
properties	O
then	O
the	O
problem	O
poses	O
efficient	B-Algorithm
polynomial-time	I-Algorithm
approximation	I-Algorithm
schemes	I-Algorithm
on	O
(	O
apex	O
)	O
minor-free	O
graphs	O
.	O
</s>
<s>
In	O
particular	O
,	O
by	O
making	O
use	O
of	O
bidimensionality	B-General_Concept
,	O
it	O
was	O
shown	O
that	O
feedback	O
vertex	O
set	O
,	O
vertex	O
cover	O
,	O
connected	O
vertex	O
cover	O
,	O
cycle	O
packing	O
,	O
diamond	O
hitting	O
set	O
,	O
maximum	O
induced	O
forest	O
,	O
maximum	O
induced	O
bipartite	O
subgraph	O
and	O
maximum	O
induced	O
planar	O
subgraph	O
admit	O
an	O
EPTAS	B-Algorithm
on	O
H-minor-free	O
graphs	O
.	O
</s>
<s>
Edge	O
dominating	O
set	O
,	O
dominating	O
set	O
,	O
r-dominating	O
set	O
,	O
connected	O
dominating	O
set	O
,	O
r-scattered	O
set	O
,	O
minimum	O
maximal	O
matching	O
,	O
independent	O
set	O
,	O
maximum	O
full-degree	O
spanning	O
tree	O
,	O
maximum	O
induced	O
at	O
most	O
d-degree	O
subgraph	O
,	O
maximum	O
internal	O
spanning	O
tree	O
,	O
induced	O
matching	O
,	O
triangle	O
packing	O
,	O
partial	O
r-dominating	O
set	O
and	O
partial	O
vertex	O
cover	O
admit	O
an	O
EPTAS	B-Algorithm
on	O
apex-minor-free	O
graphs	O
.	O
</s>
<s>
A	O
parameterized	B-General_Concept
problem	I-General_Concept
with	O
a	O
parameter	O
k	O
is	O
said	O
to	O
admit	O
a	O
linear	O
vertex	O
kernel	O
if	O
there	O
is	O
a	O
polynomial	O
time	O
reduction	O
,	O
called	O
a	O
kernelization	B-General_Concept
algorithm	O
,	O
that	O
maps	O
the	O
input	O
instance	O
to	O
an	O
equivalent	O
instance	O
with	O
at	O
most	O
O(k )	O
vertices	O
.	O
</s>
