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Heuristics for Exact 1-Planarity Testing

Authors Simon D. Fink , Miriam Münch , Matthias Pfretzschner , Ignaz Rutter

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Subject Classification

ACM Subject Classification
  • Theory of computation → Backtracking
  • Mathematics of computing → Graph algorithms
Keywords
  • 1-Planarity
  • Experiments
  • Backtracking

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Abstract

Since many real-world graphs are nonplanar, the study of graphs that allow few crossings per edge has been an active subfield of graph theory in recent years. One of the most natural generalizations of planar graphs are the so-called 1-planar graphs that admit a drawing with at most one crossing per edge. Unfortunately, testing whether a graph is 1-planar is known to be NP-complete even for very restricted graph classes. On the positive side, Binucci, Didimo and Montecchiani [Binucci et al., 2023] presented the first practical algorithm for testing 1-planarity based on an easy-to-implement backtracking strategy. We build on this idea and systematically explore the design choices of such algorithms and propose several new ingredients, such as different branching strategies and multiple filter criteria that allow us to reject certain branches in the search tree early on. We conduct an extensive experimental evaluation that evaluates the efficiency and effectiveness of these ingredients. Given a time limit of three hours per instance, our best configuration is able to solve more than 95% of the non-planar instances from the well-known North and Rome graphs with up to 50 vertices. Notably, the median running time for solved instances is well below 4 seconds.

Cite As Get BibTex

Simon D. Fink, Miriam Münch, Matthias Pfretzschner, and Ignaz Rutter. Heuristics for Exact 1-Planarity Testing. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.4

Author Details

Simon D. Fink
  • Algorithms and Complexity Group, TU Wien, Austria
Miriam Münch
  • Faculty of Computer Science and Mathematics, University of Passau, Germany
Matthias Pfretzschner
  • Faculty of Computer Science and Mathematics, University of Passau, Germany
Ignaz Rutter
  • Faculty of Computer Science and Mathematics, University of Passau, Germany

Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 541433306.
  • Fink, Simon D.: Simon D. Fink acknowledges support from Projects No. 10.47379/ICT22029 of the Vienna Science Foundation (WWTF) and No. 10.55776/Y1329 of the Austrian Science Fund (FWF).

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