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Structural Parameterizations of Simultaneous Planarity

Authors Thomas Depian , Simon D. Fink , Alexander Firbas , Robert Ganian , Matthias Pfretzschner , Ignaz Rutter

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • SEFE
  • Simultaneous Planarity
  • Fixed-Parameter Tractability
  • NP-hardness

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Abstract

Given a set of graphs on the same vertex set, the problem Simultaneous Embedding With Fixed Edges (SEFE) asks, whether there exist planar drawings of all input graphs, such that every pair of drawings coincides on their shared subgraph. It is known that SEFE is NP-complete [Elisabeth Gassner et al., 2006], even in the so-called sunflower case, where all pairs of input graphs have the same shared graph G_∩ [Marcus Schaefer, 2012]. Fink, Pfretzschner, and Rutter [Simon D. Fink et al., 2023] recently initiated the study of the parameterized complexity of SEFE in the sunflower case, mainly focusing on structural parameters of G_∩. In this work, we shift the focus towards parameters of the union graph G_∪ that contains the edges of all input graphs. On the positive side, we establish fixed-parameter tractability for the problem with respect to the feedback edge set number of G_∪. We complement this result by showing that it, surprisingly, remains NP-complete even if G_∪ has constant vertex cover number. These results settle two open questions posed by Fink et al. [Simon D. Fink et al., 2023].

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Thomas Depian, Simon D. Fink, Alexander Firbas, Robert Ganian, Matthias Pfretzschner, and Ignaz Rutter. Structural Parameterizations of Simultaneous Planarity. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.25

Author Details

Thomas Depian
  • TU Wien, Austria
Simon D. Fink
  • TU Wien, Austria
Alexander Firbas
  • TU Wien, Austria
Robert Ganian
  • TU Wien, Austria
Matthias Pfretzschner
  • University of Passau, Germany
Ignaz Rutter
  • University of Passau, Germany

Funding

Thomas Depian, Simon D. Fink, Alexander Firbas, and Robert Ganian acknowledge support from the Vienna Science and Technology Fund (WWTF) [10.47379/ICT22029]. Simon D. Fink and Robert Ganian also acknowledge support from the Austrian Science Fund (FWF) [10.55776/Y1329 and 10.55776/COE12]. Matthias Pfretzschner and Ignaz Rutter acknowledge support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [541433306].

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