By Arnold L. Rosenberg;Lenwood S. Heath
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Extra info for Graph Separators, with Applications (Frontiers in Computer Science)
Example text
If, moreover, S(n) = for some then has a recursive edge-bisector of size PROOF SKETCH. We establish the following claim by induction on The theorem will follow from the claim by direct calculation. Claim. The graph described in the statement of the theorem has a recursive bisector of size6 where We focus on a specific graph and assume, for induction, that the claim holds for all graphs having fewer than nodes. We extend the induction by laying the nodes of out on a line in a way that allows us to bisect recursively within the bounds of the claim.
By extension, two indexed families of graphs and are quasi-isometric if there is a constant c such that, for each i, the graphs 34 1 • A Technical Introduction and are c-isometric. , Rosenberg and Snyder [1978], among other sources) numerous general structural properties of graph families that preclude quasi-isometry (cf. 2), but proofs that establish quasi-isometry tend to be quite specific to the graph families in question. Most results about graph separators and graph embeddings in the literature hold only up to constant factors, hence do not distinguish between quasi-isometric families of graphs.
1983]; it is the first published instance of a cost trade-off within the world of graph embeddings. 5 are part of the folklore (although some are reviewed in Rosenberg and Snyder [1978]). 4(a) originate in Feldmann and Unger [1992]. 5(b), which is unpublished work of R. Blumofe and S. Toledo [1992]. Notes 1. For each cited area we give only a small list of seminal works, to give the reader a starting place to explore the topic. The various lists of references in the last section of each chapter supplement these lists.