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Let and let be a -ply neighborhood system

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in . The -overlap graph for the neighborhood system is the graph with vertex set

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and edge set, the subset of de¬ned by ¢

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thh ” ”

}

¤ ¢D } ) „ h 6 x C „ x }

¤ ¢ D } C „ A6 x ) „ x }

and

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Ht h ha

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A mesh in is associated with an overlap graph by assigning the coordinate of the center

3 of disk to each node of the graph. Overlap graphs model computational meshes in

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dimensions. Indeed, every mesh with bounded aspect ratio elements (ratio of largest to

smallest edge length of each element) is contained in an overlap graph. In addition, any

planar graph is an overlap graph. The main result regarding separators of overlap graphs is

the following theorem [150].

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Let be an -vertex overlap graph in dimensions. Then the

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i t

a

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vertices of can be partitioned into three sets , and such that:

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t

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No edge joins and .

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and each have at most vertices.

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o }

¡ ¡ ¡

c c

r

has only vertices.

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Thus, for , the theorem states that it is possible to partition the graph into two

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—t

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# “£§

¢

! hC §¤ !

¤ …S !

§

¥

¤ ¤

a

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subgraphs and , with a separator , such that the number of nodes for each of and

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does not exceed vertices in the worst case and such that the separator has a number

i

¨ £ c i ¨ £ c 56 }

of nodes of the order . ¢ x

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Spectral bisection refers to a technique which exploits some known properties of the eigen-

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vectors of the Laplacean of a graph. Given an adjacency graph , we associate 5t ”x

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to it a Laplacian matrix which is a sparse matrix having the same adjacency graph and

¢

de¬ned as follows: