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Monotone Clustered Level Planarity

Authors Simon D. Fink , Matthias Pfretzschner , Ignaz Rutter , Marie Diana Sieper

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Design and analysis of algorithms
  • Human-centered computing → Graph drawings
Keywords
  • Level Planarity
  • Clustered Planarity
  • Cluster Augmentation
  • Parameterized Complexity

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Abstract

We consider the combination of the two constrained planarity problems Level- and Clustered Planarity. Traditionally, level-planar drawings with convex clusters have been studied in this setting. Fink et al. [Simon D. Fink et al., 2024] recently introduced a different way of combining level- and clustered planarity by mimicking a classic characterization of clustered planarity in the level-planar setting: The problem (y-)monotone Clustered Level Planarity (mCLP) seeks a level-planar drawing in which it is possible to augment each cluster with edges that do not cross cluster boundaries so that it becomes connected while maintaining level-planarity. This is in line with previous research on clustered planarity that poses certain requirements on the augmentation edges that make each cluster connected, e.g., that they form a path. Fink et al. [Simon D. Fink et al., 2024] showed that mCLP is NP-complete even for biconnected single-source graphs and instances with a constant number of levels and clusters.
We further classify the parameterized complexity of the mCLP problem by, on the one hand, showing hardness even for instances that consist of a forest with trees of bounded size, no isolated vertices, and a small constant number of either clusters or levels. This excludes fixed-parameter tractability for almost all graph-structural parameters, except for vertex cover, even in conjunction with the number of clusters. We complement this by showing fixed-parameter tractability when parameterizing by the vertex cover number and the number of clusters. A major obstacle that we overcome is the fact that mCLP is non-hereditary, i.e., subinstances of yes-instances may be no-instances and vice versa, which makes it challenging to apply usual reduction techniques.

Cite As Get BibTex

Simon D. Fink, Matthias Pfretzschner, Ignaz Rutter, and Marie Diana Sieper. Monotone Clustered Level Planarity. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.14

Author Details

Simon D. Fink
  • Algorithms and Complexity Group, Technische Universität Wien, Austria
Matthias Pfretzschner
  • Faculty of Computer Science and Mathematics, University of Passau, Germany
Ignaz Rutter
  • Faculty of Computer Science and Mathematics, University of Passau, Germany
Marie Diana Sieper
  • Institute of Computer Science, University of Würzburg, Germany

References

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