Embedding parameterized graph classes into normed spaces

The central result in the theory of metric embeddings is Bourgain's [7] result that an arbitrary metric on n points can be embedded into ℓp for p 1 with O(log n) distortion. Since Linial et al. [29] showed this bound to be tight, there has been great interest in embeddings of restricted classes of metrics.A primary area of interest has been the embedding of graph metrics into normed spaces. Such embeddings allow geometric methods to be applied to problems defined on graphs. One of the deepest conjectures in this area is that graphs which exclude a particular minor can be embedded into ℓ1 with distortion that depends only on that minor. If true, this would have the following important consequence: Treewidth-k graphs can be embedded into ℓ 1 with distortion O(f (k)) Motivated by this, I consider the more general question: "For any p, is there a graph parameter such that when the parameter has value k we can embed a graph into ℓ p with distortion O(f ( k))?".In this dissertation I answer this question in the affirmative and I make the following contributions: (1) I introduce a new technique called iterative embedding by which local embeddings can be combined to form a global embedding. (2) I formulate a new graph parameter called tree-bandwidth which is similar to treewidth and use iterative embedding to show that tree-bandwidth-k graphs can be embedded into ℓ1 with distortion O( k3 log k). This is the first constant distortion embedding of a non-trivial, non-planar graph family into ℓ 1. (3) I prove that bandwidth-k graphs can be embedded with distortion O(k) into graphs with tree-bandwidth-k. This gives O( k4 log k) distortion ℓ1 -embeddings for bandwidth-k graphs. (4) Furthermore, for bandwidth-k graphs, these techniques give O( k4 log k) distortion l p-embeddings for p ≥ 1. This is the first constant distortion embedding of an infinite graph class into ℓ2 . (5) I prove lower bounds on the distortion of probabilistic embeddings of minor excluded graphs into graphs which exclude simpler minors. Note that this rules out a plausible approach to embedding treewidth-k graphs into ℓ1.