Randomly removing g handles at once
Glencora Borradaile, James R. Lee, Anastasios Sidiropoulos

TL;DR
This paper introduces a method to embed graphs of genus g into planar graphs with quadratic distortion by removing all handles simultaneously, improving upon previous iterative approaches.
Contribution
It presents a novel approach for embedding genus g graphs into planar graphs with O(g^2) distortion by removing all handles at once, using the Peeling Lemma.
Findings
Achieves O(g^2) distortion embedding for genus g graphs.
Utilizes the low dilation property of the minimum-cut graph.
Employs the Peeling Lemma to randomly cut the surface.
Abstract
Indyk and Sidiropoulos (2007) proved that any orientable graph of genus can be probabilistically embedded into a graph of genus with constant distortion. Viewing a graph of genus as embedded on the surface of a sphere with handles attached, Indyk and Sidiropoulos' method gives an embedding into a distribution over planar graphs with distortion , by iteratively removing the handles. By removing all handles at once, we present a probabilistic embedding with distortion for both orientable and non-orientable graphs. Our result is obtained by showing that the nimum-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).
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Complexity and Algorithms in Graphs · Advanced Graph Theory Research
