Computational Topology (Jeff Erickson)
Graph Separators
In the spring of 1930,. . . König told me that he was about to finish a book that would include
all that was known about graphs. I assured him that such a book would fill a great need; and I
brought up my
n
Arc Theorem which, having been published as a lemma in a curvetheoretical
paper, had not yet come to his attention. König was greatly interested, but did not believe
the theorem was correct. “This evening,” he said to me in parting, “I won’t go to sleep before
having constructed a counterexample!”. When me met again the next day he greeted me
with the words “Sleepless night!”
— Karl Menger, “On the origin of the
n
arc theorem”,
J. Graph Theory
5:341–350, 1981.
10
Graph Separators
“Divide and conquer” is one of the oldest and most widely used techniques for designing efficient algo
rithms. Divideandconquer algorithms partition their inputs into two or more independent subproblems,
solve those subproblems recursively, and then combine the solutions to those subproblems to obtain
their final output. This strategy can be successfully applied to several graph problems, provided we can
quickly
separate
the graph into roughly equal subgraphs.
An
"
separator
of an
n
vertex graph
G
= (
V
,
E
)
is a subset
S
⊆
V
such that each connected component
of
G
\
S
has at most
"
n
vertices. Our goal is to find
"
separators, for some constant 1
/
2
≤
" <
1, that
have few vertices. For example, any path has a 1
/
2separator consisting of a single vertex; any binary
tree has a 2
/
3separator consisting of a single vertex; and any outerplanar graph has a 2
/
3separator
consisting of two vertices.
The following classical theorem of Menger
[
12
]
, which is both a precursor and an easy consequence
of the maxflowmincut theorem, is a key tool in proving the existence of small separators.
Theorem 10.1 (Menger).
Let
G
= (
V
,
E
)
be a graph. The minimum number of vertices separating any
subsets
A
,
b
⊆
V
is equal to the maximum number of vertexdisjoint paths from
A
to
B
.
10.1
Planar Separators
In the late 1970s, Richard Lipton and Robert Tarjan
[
11
]
proved the following seminal result.
The Planar Separator Theorem.
Any
n
vertex planar graph has a
2
/
3
separator containing at most
p
8
n
vertices.
Proof (Alon, Seymour, and Thomas
[
2
]
):
Let
G
be an embedded planar graph with
n
≥
3 vertices,
and let
k
=
b
p
2
n
c
. Without loss of generality, we can assume that
G
has no loops or parallel edges,
and that every face is a triangle bounded by three distinct edges. For any simple cycle
C
in
G
, let
In
(
C
)
and
Out
(
C
)
denote the vertices inside and outside
C
, respectively. No vertex of
In
(
C
)
is adjacent to any
vertex of
Out
(
C
)
. Let
C
be a simple cycle satisfying three conditions:
(1)
C
has at most 2
k
vertices.
(2)

Out(C)

<
2
n
/
3.
(3) Subject to conditions (1) and (2), the difference

In
(
C
)
  
Out
(
C
)

is as small as possible.
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 Fall '08
 Staff
 Graph Theory, vertices

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