Synchronized Planarity with Applications to Constrained Planarity Problems

Abstract

We introduce the problem S ynchronized P lanarity . 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. S ynchronized P lanarity 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 S ynchronized P lanarity 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 S ynchronized P lanarity . In particular, this lets us solve C lustered P lanarity in quadratic time, where the most efficient previously known algorithm has an upper bound of O ( n 8 ).