A Planarity Test via Construction Sequences

TL;DR

This paper introduces a simplified linear-time planarity testing algorithm that maintains a 3-connected planar embedding throughout the process, providing a new approach distinct from traditional methods.

Contribution

It presents a novel reduction from planarity testing to constructing a 3-connected graph, simplifying the process while ensuring linear runtime.

Findings

Algorithm runs in linear time

Successfully computes planar embeddings for planar graphs

Identifies Kuratowski subdivisions in non-planar graphs

Abstract

Optimal linear-time algorithms for testing the planarity of a graph are well-known for over 35 years. However, these algorithms are quite involved and recent publications still try to give simpler linear-time tests. We give a simple reduction from planarity testing to the problem of computing a certain construction of a 3-connected graph. The approach is different from previous planarity tests; as key concept, we maintain a planar embedding that is 3-connected at each point in time. The algorithm runs in linear time and computes a planar embedding if the input graph is planar and a Kuratowski-subdivision otherwise.