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By Arnold L. Rosenberg;Lenwood S. Heath

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We claim that this node-assignment witnesses the claimed subgraph relation. Note first that, because this assignment is single-valued and onto, it must also be one-to-one. 6-2. Illustrating the shuffle-exchange graph as a subgraph of the (undirected) de Bruijn graph; nodes are depicted by (name in shuffle-exchange)/(name in de Bruijn). 6 • Quasi-Isometric Graph Families 37 Shuffle adjacencies. Each shuffle-edge of connects nodes of equal weights; therefore, the indicated node-assignment preserves shuffle adjacencies in Exchange adjacencies.

Four fundamental measures of the quality of a graph embedding have proven important in the many applications of embeddings. We focus primarily on these measures throughout the book, although we do present here also a fifth measure that has more limited application. The reader should be aware that a variety of other, special-purpose, measures are useful also in particular studies, as we shall see in Chapter 2. In order to define the cost measures of interest, let us focus on an embedding of the graph into the graph We begin with the four primary cost measures.

4. 4-2. A typical internal node x and its children in a decomposition tree. , Fr )-decomposition tree. ,1)-decomposition tree for , in the sense that either the bifurcator or the decomposition tree can easily be transformed into the other. Our goal here is to show that an arbitrary decomposition tree for can be transformed into a fully balanced one that has small edge imbalance. Before doing so, it is convenient to digress into the somewhat unexpected realm of strings of pearls. 2. 1a. Bisecting a String of Pearls.

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