Fully-dynamic Planarity Testing in Polylogarithmic Time

TL;DR

This paper introduces a deterministic fully-dynamic algorithm for maintaining graph planarity with polylogarithmic update time, significantly improving efficiency over previous methods for dynamic graph planarity testing.

Contribution

It presents the first deterministic fully-dynamic planarity testing algorithm with amortized $O( ext{log}^3 n)$ time per update for general graphs.

Findings

Achieves $O( ext{log}^3 n)$ amortized update time for planarity testing.

Maintains a planarity indicator bit efficiently during dynamic updates.

Provides exponential improvement over previous $O( ext{sqrt} n)$ algorithms.

Abstract

Given a dynamic graph subject to insertions and deletions of edges, a natural question is whether the graph presently admits a planar embedding. We give a deterministic fully-dynamic algorithm for general graphs, running in amortized $O(\log^3 n)$ time per edge insertion or deletion, that maintains a bit indicating whether or not the graph is presently planar. This is an exponential improvement over the previous best algorithm [Eppstein, Galil, Italiano, Spencer, 1996] which spends amortized $O(\sqrt{n})$ time per update.