Computational Topology (Jeff Erickson)
Treewidth
Or il y avait des graines terribles sur la planète du petit prince . . . c’étaient les graines
de baobabs. Le sol de la planète en était infesté. Or un baobab, si l’on s’y prend trop
tard, on ne peut jamais plus s’en débarrasser. Il encombre toute la planète. Il la perfore
de ses racines. Et si la planète est trop petite, et si les baobabs sont trop nombreux, ils
la font éclater.
[Now there were some terrible seeds on the planet that was the home of the little
prince; and these were the seeds of the baobab. The soil of that planet was infested with
them. A baobab is something you will never, never be able to get rid of if you attend to
it too late. It spreads over the entire planet. It bores clear through it with its roots. And
if the planet is too small, and the baobabs are too many, they split it in pieces.]
— Antoine de SaintExupéry (translated by Katherine Woods)
Le Petit Prince [The Little Prince]
(1943)
11
Treewidth
In this lecture, I will introduce a graph parameter called
treewidth
, that generalizes the property of
having small separators. Intuitively, a graph has small treewidth if it can be
recursively
decomposed into
small subgraphs that have small overlap, or even more intuitively, if the graph resembles a ‘fat tree’.
Many problems that are NPhard for general graphs can be solved in polynomial time for graphs with
small treewidth. Graphs embedded on surfaces of small genus do not necessarily have small treewidth,
but they can be covered by a small number of subgraphs, each with small treewidth. This covering can
be used to develop efficient algorithms (either exact or approximate) for a huge number of problems
on surface graphs that are NPhard for general graphs. As an example of this technique, I’ll describe a
polynomialtime approximation scheme for the maximum independent set problem.
11.1
Definitions
The concept of treewidth was discovered independently by several different researchers and given
several different names. The actual term ‘treewidth’ and its definition in terms of tree decompositions
were introduced by Robertson and Seymour
[
15
]
. Almost simultaneously, Arnborg and Proskurowski
independently began the systematic study of
partial
k
trees
[
3
,
2
,
1
]
. Both groups were apparently
unaware of Bertelè and Brioschi’s earlier equivalent definition of the
dimension
1
of a graph
[
6
]
, or
Halin’s related study of
S
functions
[
13
]
. Other equivalent definitions are surveyed by Bodlaender
[
8
,
9
]
.
A
tree decomposition
(
T
,
X
)
of a graph
G
= (
V
,
E
)
consists of a tree
T
= (
I
,
F
)
and a function
X
:
I
→
2
V
satisfying three constraints. To avoid confusion, I will always refer to
vertices
of
G
, but
nodes
of
T
. We say that a vertex
v
is
associated with
a node
i
, or vice versa, whenever
v
∈
X
(
i
)
.
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 Fall '08
 Staff
 Graph Theory, The Little Prince, planar graphs, tree decomposition, largest independent set

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