CHAPTER 2
Isoperimetric problems
2.1. History
One of the earliest problems in geometry was the isoperimetric problem, which
was considered by the ancient Greeks.
The problem is to find, among all closed
curves of a given length, the one which encloses the maximum area.
The basic
isoperimetric problem for graphs is essentially the same. Namely, remove as little
of the graph as possible to separate out a subset of vertices of some desired “size”.
Here the size of a subset of vertices may mean the number of vertices, the number
of edges, or some other appropriate measure defined on graphs.
A typical case
is to remove as few edges as possible to disconnect the graph into two parts of
almost equal size.
Such problems are usually called
separator
problems and are
particularly useful in a number of areas including recursive algorithms, network
design, and parallel architectures for computers, for example [
183
].
In a graph, a subset of edges which disconnects the graph is called a
cut
.
Cuts arise naturally in the study of connectivity of graphs where the sizes of the
disconnected parts are not of concern.
Isoperimetric problems examine optimal
relations between the size of the cut and the sizes of the separated parts.
Many
different names are used for various versions of isoperimetric problems (such as the
conductance of a graph, the isoperimetric number, etc.). The concepts are all quite
similar, but the differences are due to the varying definitions of cuts and sizes.
We will consider two types of cuts. A
vertexcut
is a subset of vertices whose
removal disconnects the graph. Similarly, an
edgecut
is a subset of edges whose
removal separates the graph. The size of a subset of vertices depends on either the
number of vertices or the number of edges. Therefore, there are several combina
tions.
Roughly speaking, isoperimetric problems involving edgecuts correspond in a
natural way to Cheeger constants in spectral geometry. The formulation and the
proof techniques are very similar. Cheeger constants were studied in the thesis of
Cheeger [
52
], but they can be traced back to Poly´
a and Szeg¨
o [
216
]. We will follow
tradition and call the discrete versions by the same names, such as the Cheeger
constant and the Cheeger inequalities.
23
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2. ISOPERIMETRIC PROBLEMS
2.2. The Cheeger constant of a graph
Before we discuss isoperimetric problems for graphs, let us first consider a
measure on subsets of vertices. The typical measure assigns weight 1 to each vertex,
so the measure of a subset is its number of vertices. However, this implies that all
vertices have the same measure. For some problems, this is appropriate only for
regular graphs and does not work for general graphs. The measure we will use here
takes into consideration the degree of a vertex. For a subset
S
of the vertices of
G
,
we define vol
S
, the
volume
of
S
, to be the sum of the degrees of the vertices in
S
:
vol
S
=
X
x
∈
S
d
x
,
for
S
⊆
V
(
G
).
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 Spring '08
 aterras
 Math, Geometry, Graph Theory, Inequalities, Cartesian product, Expander graph, Isoperimetric inequality

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