Randomly removing g handles at once

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 $g$ 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 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).