We introduce the problem Synchronized Planarity. Roughly speaking, its input is a loop-free multi-graph together with synchronization constraints that, e.g., match pairs of vertices of equal degree by providing a bijection between their edges. Synchronized Planarity then asks whether the graph admits a crossing-free embedding into the plane such that the orders of edges around synchronized vertices are consistent. We show, on the one hand, that Synchronized Planarity can be solved in quadratic time, and, on the other hand, that it serves as a powerful modeling language that lets us easily formulate several constrained planarity problems as instances of Synchronized Planarity. In particular, this lets us solve Clustered Planarity in quadratic time, where the most efficient previously known algorithm has an upper bound of O(n⁸).

我们引入同步平面性问题。粗略地说，它的输入是一个无循环多重图以及同步约束，例如，通过在边缘之间提供双射来匹配相等度数的顶点对。然后，同步平面性询问图是否允许无交叉嵌入到平面中，以使同步顶点周围的边的顺序一致。一方面，我们证明了同步平面性可以在二次时间内解决，另一方面，它可以作为一种强大的建模语言，使我们可以轻松地将几个约束平面性问题表述为同步平面性的实例平面度。特别是，这让我们能够在二次时间内解决 C lustered P lanarity，其中最有效的先前已知算法的上限为O ( n ⁸ )。