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Exact Algorithms for Clustered Planarity with Linear Saturators

Authors Giordano Da Lozzo , Robert Ganian , Siddharth Gupta , Bojan Mohar , Sebastian Ordyniak , Meirav Zehavi

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Author Details

Giordano Da Lozzo
  • Roma Tre University, Italy
Robert Ganian
  • Algorithms and Complexity Group, TU Wien, Austria
Siddharth Gupta
  • BITS Pilani, K K Birla Goa Campus, India
Bojan Mohar
  • Department of Mathematics, Simon Fraser University, Burnaby, Canada
Sebastian Ordyniak
  • University of Leeds, UK
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Da Lozzo, Giordano: Supported, in part, by MUR of Italy (PRIN Project no. 2022ME9Z78 - NextGRAAL and PRIN Project no. 2022TS4Y3N - EXPAND).
  • Ganian, Robert: Supported by the Austrian Science Fund (FWF, Project 10.55776/Y1329) and the Vienna Science and Technology Fund (WWTF, Project 10.47379/ICT22029).
  • Gupta, Siddharth: Supported, in part, by BITS Pilani New Faculty Seed Grant.
  • Mohar, Bojan: Supported, in part, by the NSERC Discovery Grant R832714 (Canada), by the ERC Synergy grant (European Union, ERC, KARST, project number 101071836), and by the Research Project N1-0218 of ARIS (Slovenia).
  • Zehavi, Meirav: Supported by the European Research Council (ERC) grant titled PARAPATH.

Acknowledgements

This research started at the Dagstuhl Seminar: New Frontiers of Parameterized Complexity in Graph Drawing; seminar number: 23162; April 16-21, 2023.

Cite As Get BibTex

Giordano Da Lozzo, Robert Ganian, Siddharth Gupta, Bojan Mohar, Sebastian Ordyniak, and Meirav Zehavi. Exact Algorithms for Clustered Planarity with Linear Saturators. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.24

Abstract

We study Clustered Planarity with Linear Saturators, which is the problem of augmenting an n-vertex planar graph whose vertices are partitioned into independent sets (called clusters) with paths - one for each cluster - that connect all the vertices in each cluster while maintaining planarity. We show that the problem can be solved in time 2^𝒪(n) for both the variable and fixed embedding case. Moreover, we show that it can be solved in subexponential time 2^𝒪(√n log n) in the fixed embedding case if additionally the input graph is connected. The latter time complexity is tight under the Exponential-Time Hypothesis. We also show that n can be replaced with the vertex cover number of the input graph by providing a linear (resp. polynomial) kernel for the variable-embedding (resp. fixed-embedding) case; these results contrast the NP-hardness of the problem on graphs of bounded treewidth (and even on trees). Finally, we complement known lower bounds for the problem by showing that Clustered Planarity with Linear Saturators is NP-hard even when the number of clusters is at most 3, thus excluding the algorithmic use of the number of clusters as a parameter.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph algorithms
Keywords
  • Clustered planarity
  • independent c-graphs
  • path saturation
  • graph drawing

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