Synchronized Planarity with Applications to Constrained Planarity Problems

TL;DR

This paper introduces Synchronized Planarity, a new problem that efficiently models and solves various constrained planarity problems, including Clustered Planarity, with quadratic time algorithms.

Contribution

It defines Synchronized Planarity, proves its quadratic-time solvability, and demonstrates its effectiveness as a modeling framework for constrained planarity problems.

Findings

Synchronized Planarity can be solved in quadratic time.

It provides a new modeling approach for constrained planarity problems.

Enables quadratic-time solutions for Clustered Planarity.

Abstract

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 $O(n^{8})$.