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Unformatted text preview: 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 vertex-cut is a subset of vertices whose removal disconnects the graph. Similarly, an edge-cut 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 edge-cuts 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 24 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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