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].

翻译：平面嵌入是指将图画在平面上，使得边除顶点外互不相交。已知当整个图作为输入时，无论是顺序算法[HT]还是并行算法[RR94]，都能高效地测试图的平面性并计算其嵌入（若存在）。在动态场景下，输入图通过边插入和删除每次改变一条边，平面性测试/嵌入需在每次变化后更新。通过存储辅助信息，我们可以改善动态平面性测试/嵌入的复杂度，避免明显的从头重新计算。在顺序动态场景中，已有系列研究[EGIS, IPR, HIKLR, HR1]，最终以Holm与Rotenberg[HR2]提出的多对数级顺序时间（摊销）平面性测试算法作为突破性成果。本文通过DynFO（Patnaik等人[PI]（亦见[DST95]）引入的并行动态复杂度类）的视角研究平面嵌入。我们证明：在DynFO中，可以动态维护一条边能否插入平面图而不破坏平面性。进一步地，我们展示了如何在边插入和删除操作下维护平面图的嵌入，同时拒绝违反平面性的边插入。核心思路是仅维护三连通分量的嵌入以及分隔对的一种特殊二染色，这使我们能够规避双连通平面图嵌入变化时的级联翻转问题——该问题是顺序动态算法[HR1, HR2]的主要难点。