Planarizing an Unknown Surface

TL;DR

This paper presents a method to embed graphs of small genus into planar graphs without requiring a known drawing, enabling solutions for genus-related problems without prior surface representations.

Contribution

It introduces a novel approach to compute embeddings of genus-g graphs into planar graphs without needing a surface drawing, expanding applicability.

Findings

Embedding can be computed without a known surface drawing

Reduces problems on small genus graphs to planar graph problems

First known method for such genus-to-planar embedding without surface input

Abstract

It has been recently shown that any graph of genus g>0 can be stochastically embedded into a distribution over planar graphs, with distortion Olog (g+1)) [Sidiropoulos, FOCS 2010]. This embedding can be computed in polynomial time, provided that a drawing of the input graph into a genus-g surface is given. We show how to compute the above embedding without having such a drawing. This implies a general reduction for solving problems on graphs of small genus, even when the drawing into a small genus surface is unknown. To the best of our knowledge, this is the first result of this type.