On the Geometry of Graphs with a Forbidden Minor James R. Lee ∗ University of Washington jrl@cs.washington.edu Anastasios Sidiropoulos University of Toronto tasoss@cs.toronto.edu ABSTRACT We study the topological simplification of graphs via ran- dom embeddings, leading ultimately to a reduction of the Gupta-Newman-Rabinovich-Sinclair (GNRS) L1 embedding conjecture to a pair of manifestly simpler conjectures. The GNRS conjecture characterizes all graphs that have an O(1)- approximate multi-commodity max-flow/min-cut theorem. In particular, its resolution would imply a constant factor approximation for the general Sparsest Cut problem in ev- ery family of graphs which forbids some minor. In the course of our study, we prove a number of results of independent interest. • Every metric on a graph of pathwidth k embeds into a distribution over trees with distortion depending only on k. This is equivalent to the statement that any fam- ily of graphs excluding a fixed tree embeds into a dis- tribution over trees with O(1) distortion. For graphs of treewidth k, GNRS showed that this is impossible even for k = 2. In particular, our result implies that pathwidth-k met- rics embed into L 1 with bounded distortion, which re- solves an open question even for k = 3. • We prove a generic peeling lemma which uses random retractions to peel simple structures like handles and apices off of graphs. This allows a number of new topological reductions. For example, if X is any met- ric space in which the removal of O(1) points leaves a bounded genus metric, then X embeds into a distribu- tion over planar graphs. • Using these techniques, we show that the GNRS em- bedding conjecture is equivalent to two simpler con- jectures: (1) The well-known planar embedding con- jecture, and (2) a conjecture about embeddings of k- sums of graphs. ∗ Research supported by NSF CAREER award CCF- 0644037. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. STOC’09, May 31–June 2, 2009, Bethesda, Maryland, USA. Copyright 2009 ACM 978-1-60558-506-2/09/05 ...$5.00. Categories and Subject Descriptors F.2 [Analysis of Algorithms and Problem Complex- ity]: Miscellaneous General Terms Algorithms, Theory Keywords Embeddings, Geometry of Graphs, Forbidden Minors 1. INTRODUCTION We view an undirected graph G =(V,E) as a topologi- cal template that supports a number of different geometries. Such a geometry is specified by a non-negative length func- tion len : E → R on edges, which induces a shortest-path pseudometric d len on V , with d len (u, v) = length of the shortest path between u and v, where a pseudometric might have d len (u, v) = 0 for some pairs u, v ∈ V with u = v. From this point of view, we are interested in properties which hold simultaneously for all geometries supported on G, or even for all geometries supported on a family of graphs F . In the seminal works of Linial, London, and Rabinovich [19], and later Aumann and Rabani [1] and Gupta, New- man, Rabinovich, and Sinclair [11], the geometry of graphs is related to the classical study of the relationship between flows and cuts. Multi-commodity flows and L1 embeddings. For any metric space (X, d), we use c 1 (X, d) to denote the L 1 dis- tortion of (X, d), i.e. the infimum over all numbers D such that X admits an embedding f : X → L1 with d(x, y) ≤‖f (x) − f (y)‖1 ≤ D · d(x, y) for all x, y ∈ X. Here, we have L 1 = L 1 ([0, 1]), which can be replaced by the sequence space ℓ 1 when X is finite. Corresponding to the preceding discussion, for a graph G =(V,E) we write c1(G) = sup c1(V,d) where d ranges over all metrics supported on G, and for a family F of graphs, we write c 1 (F ) = sup G∈F c 1 (G). Thus for a family F of finite graphs, c 1 (F ) ≤ D if and only if every geometry supported on a graph in F embeds into L 1 with distortion at most D. On the other hand, one has the notion of a multi-commodity flow instance in G which is specified by a pair of non-negative mappings cap : E → R and dem : V × V → R. We write 245