Consider two planar graphs which are subject to edge insertions and deletions. We show that whether the two graphs are isomorphic can be maintained with first-order logic formulas and auxiliary data of polynomial size. This places the dynamic planar graph isomorphism problem into the dynamic descriptive complexity class DynFO. As a consequence, there is a dynamic constant-time parallel algorithm with polynomial-size auxiliary data which maintains whether two dynamic planar graphs are isomorphic.

翻译：考虑两个受到边插入与删除操作的平面图。我们证明，两个图是否同构可以通过一阶逻辑公式及多项式大小的辅助数据来维护。这一结果将动态平面图同构问题归入动态描述复杂性类DynFO。因此，存在一种具有多项式大小辅助数据的动态常量时间并行算法，能够维护两个动态平面图是否同构。