Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Lectures on Discrete Geometry
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
A tight bound on approximating arbitrary metrics by tree metrics
Journal of Computer and System Sciences - Special issue: STOC 2003
Probabilistic embeddings of bounded genus graphs into planar graphs
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Exploiting planarity for network flow and connectivity problems
Exploiting planarity for network flow and connectivity problems
On the geometry of graphs with a forbidden minor
Proceedings of the forty-first annual ACM symposium on Theory of computing
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Indyk and Sidiropoulos (2007) proved that any orientable graph of genus g can be probabilistically embedded into a graph of genus g-1 with constant distortion. Viewing a graph of genus g as embedded on the surface of a sphere with g handles attached, Indyk and Sidiropoulos' method gives an embedding into a distribution over planar graphs with distortion 2^O^(^g^), by iteratively removing the handles. By removing all g handles at once, we present a probabilistic embedding with distortion O(g^2) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of Erickson and Har-Peled (2004) has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma of Lee and Sidiropoulos (2009).