同步平面性及其在约束平面性问题中的应用
Synchronized Planarity with Applications to Constrained Planarity Problems
摘要：
我们介绍了同步平面性问题。粗略地说，它的输入是一个无环多重图以及同步约束，例如，通过在顶点的边之间提供双射来匹配度数相等的顶点对。同步平面性然后询问图是否允许无交叉嵌入到平面中，使得同步顶点周围的边的顺序是一致的。我们表明，一方面，同步平面性可以在二次时间内解决，另一方面，它作为一个强大的建模语言，让我们很容易制定几个约束平面性问题的同步平面性的实例。特别是，这使我们能够在二次时间内求解离散平面性，其中最有效的先前已知算法的上限为$O(n^{8})$。
英文摘要：
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 $O(n^{8})$.
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