Dynamic Planar Embedding is in DynFO

TL;DR

This paper demonstrates that maintaining the planarity and embedding of a dynamic graph can be achieved within the parallel dynamic complexity class DynFO, enabling efficient updates after edge insertions and deletions.

Contribution

It shows how to dynamically maintain planarity and embeddings of graphs in DynFO, extending previous work to a parallel dynamic setting and avoiding cascading updates.

Findings

Planarity testing can be maintained in DynFO during dynamic updates.

Embedding of planar graphs can be dynamically maintained in DynFO.

The approach avoids cascading flips by maintaining embeddings of triconnected components.

Abstract

Planar Embedding is a drawing of a graph on the plane such that the edges do not intersect each other except at the vertices. We know that testing the planarity of a graph and computing its embedding (if it exists), can efficiently be computed, both sequentially [HT] and in parallel [RR94], when the entire graph is presented as input. In the dynamic setting, the input graph changes one edge at a time through insertion and deletions and planarity testing/embedding has to be updated after every change. By storing auxilliary information we can improve the complexity of dynamic planarity testing/embedding over the obvious recomputation from scratch. In the sequential dynamic setting, there has been a series of works [EGIS, IPR, HIKLR, HR1], culminating in the breakthrough result of polylog(n) sequential time (amortized) planarity testing algorithm of Holm and Rotenberg [HR2]. In this paper, we study planar embedding through the lens of DynFO, a parallel dynamic complexity class introduced by Patnaik et al. [PI] (also [DST95]). We show that it is possible to dynamically maintain whether an edge can be inserted to a planar graph without causing non-planarity in DynFO. We extend this to show how to maintain an embedding of a planar graph under both edge insertions and deletions, while rejecting edge insertions that violate planarity. Our main idea is to maintain embeddings of only the triconnected components and a special two-colouring of separating pairs that enables us to side-step cascading flips when embedding of a biconnected planar graph changes, a major issue for sequential dynamic algorithms [HR1, HR2].