Planarity Testing Revisited

TL;DR

This paper introduces a simple, logspace algorithm for planarity testing and embedding of graphs, along with a method to find Kuratowski minors in non-planar graphs, improving efficiency and potential for generalization.

Contribution

It presents the first logspace algorithms for planar embedding and obstacle detection, simplifying previous complex methods and enabling future extensions.

Findings

Logspace algorithm for planar embedding

Logspace algorithm for detecting Kuratowski minors

Potential for generalization to bounded genus graphs

Abstract

Planarity Testing is the problem of determining whether a given graph is planar while planar embedding is the corresponding construction problem. The bounded space complexity of these problems has been determined to be exactly Logspace by Allender and Mahajan with the aid of Reingold's result. Unfortunately, the algorithm is quite daunting and generalizing it to say, the bounded genus case seems a tall order. In this work, we present a simple planar embedding algorithm running in logspace. We hope this algorithm will be more amenable to generalization. The algorithm is based on the fact that 3-connected planar graphs have a unique embedding, a variant of Tutte's criterion on conflict graphs of cycles and an explicit change of cycle basis.% for planar graphs. We also present a logspace algorithm to find obstacles to planarity, viz. a Kuratowski minor, if the graph is non-planar. To the best of our knowledge this is the first logspace algorithm for this problem.