6.889 — Lecture 3: Planar Separators
Christian Sommer
[email protected]
September 13, 2011
We shall prove theorems of the following ﬂavor (see textbook/papers for precise statements and proofs).
Thm.
For any planar graph
G
= (
V,E
)
on
n
=

V

vertices and for any
1
weight function
w
:
V
→
R
+
,
we can partition
V
into
A,B,S
⊆
V
such that
•
[
α
–balanced]
w
(
A
)
,w
(
B
)
≤
α
·
w
(
V
)
for some
α
∈
(0
,
1)
•
[separation]
no edge between any
a
∈
A
and
b
∈
B
(
A
×
B
∩
E
=
∅
)
•
[small separator]

S
 ≤
f
(
n
)
•
[efﬁcient]
A,B,S
can be found in linear time.
Trees
what if
G
is a (binary) tree? can do
1
/
2
–balanced partition with

S

= 1
?
only
2
/
3
–balanced!
with one
edge
in separator?
only
3
/
4
–balanced for binary trees
1
/
3
1
/
3
1
/
3
1
/
4
1
/
4
1
/
4
1
/
4
Grids
What happens for a grid on
n
vertices (say square:
√
n
×
√
n
)?

S
 ≤
√
n
1
/
2
–balanced
√
n
2
/
3
–balanced
<
√
n
but
Θ(
√
n
)
cut out a diagonal and remain
2
/
3
–balanced,
s
vertices separate
≈
s
2
/
2
vertices from the rest,
n/
3
≤
s
2
/
2
.
O
(
√
n
)
is “right order of magnitude”
today’s lecture: can generalize to
all
planar graphs
Beyond
Extensions to boundedgenus and minorfree graphs to be discussed in Lecture 5
General Graphs
Is there a separator theorem that works for
any
graph? No (complete graph)! For any
sparse graph? No (expander graphs)!
1
almost — need individual weights
≤
(1

α
)
w
(
V
)
1
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We shall prove two versions of the main theorem. Both proofs use the following lemma (a weighted version).
Lemma.
For any planar graph
G
= (
V,E
)
with a spanning tree of radius
d
rooted at
r
∈
V
,
we can partition
V
into
A,B,S
⊆
V
such that
•
[balanced]

A

,

B
 ≤
3
n/
4
•
[separation]
no edge between any
a
∈
A
and
b
∈
B
(
A
×
B
∩
E
=
∅
)
•
[separator size]

S
 ≤
2
d
+ 1
•
[efﬁcient]
A,B,S
can be found in linear time.
Proof (sketch).
Let
T
be the spanning tree of depth
d
rooted at
r
. Triangulate
G
. Recall
interdigitating trees
from Lecture 2. Let
T
*
be the dual tree in the triangulated version of
G
. Every
nontree edge
e
deﬁnes a
fundamental cycle
C
(
e
)
. Since
T
has depth
d
, we have

C
(
e
)
 ≤
2
d
+ 1
.
assign appropriate weights to faces. then ﬁnd edge separator in interdigitating tree! (
T
*
has degree
3
)
Problem Set: how to
efﬁciently
ﬁnd the best edge
e
(how to compute
w
(
ext
(
C
(
e
)))
,w
(
int
(
C
(
e
)))
for each
edge
e
, where ext
(
C
)
,
int
(
C
)
denote the
exterior
and
interior
of a cycle
C
, respectively)
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 Fall '11
 ErikDemaine
 Algorithms, Graph Theory, li, planar graphs, Richard J. Lipton

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