Synchronized Planarity with Applications to Constrained Planarity Problems

Authors Thomas Bläsius, Simon D. Fink , Ignaz Rutter

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Author Details

Thomas Bläsius
  • Faculty of Informatics, Karlsruhe Institute of Technology (KIT), Germany
Simon D. Fink
  • Faculty of Informatics and Mathematics, Universität Passau, Germany
Ignaz Rutter
  • Faculty of Informatics and Mathematics, Universität Passau, Germany

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Thomas Bläsius, Simon D. Fink, and Ignaz Rutter. Synchronized Planarity with Applications to Constrained Planarity Problems. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graphs and surfaces
  • Planarity Testing
  • Constrained Planarity
  • Cluster Planarity
  • Atomic Embeddability


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