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The Price of Connectivity Augmentation on Planar Graphs

Authors Hugo A. Akitaya , Justin Dallant , Erik D. Demaine , Michael Kaufmann , Linda Kleist , Frederick Stock , Csaba D. Tóth , Torsten Ueckerdt

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ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Computational geometry
Keywords
  • connectivity augmentation
  • local crossing number
  • flip distance

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Abstract

Given two classes of graphs, 𝒢₁ ⊆ 𝒢₂, and a c-connected graph G ∈ 𝒢₁, we wish to augment G with a smallest cardinality set of new edges F to obtain a k-connected graph G' = (V,E∪ F) ∈ 𝒢₂. In general, this is the c → k connectivity augmentation problem. Previous research considered variants where 𝒢₁ = 𝒢₂ is the class of planar graphs, plane graphs, or planar straight-line graphs. In all three settings, we prove that the c → k augmentation problem is NP-complete when 2 ≤ c < k ≤ 5. 
However, the connectivity of the augmented graph G' is at most 5 if 𝒢₂ is limited to planar graphs. We initiate the study of the c → k connectivity augmentation problem for arbitrary k ∈ ℕ, where 𝒢₁ is the class of planar graphs, plane graphs, or planar straight-line graphs, and 𝒢₂ is a beyond-planar class of graphs: 𝓁-planar, 𝓁-plane topological, or 𝓁-plane geometric graphs. We obtain tight bounds on the tradeoffs between the desired connectivity k and the local crossing number 𝓁 of the augmented graph G'. We also show that our hardness results apply to this setting.
The connectivity augmentation problem for triangulations is intimately related to edge flips; and the minimum augmentation problem to the flip distance between triangulations. We prove that it is NP-complete to find the minimum flip distance between a given triangulation and a 4-connected triangulation, settling an open problem posed in 2014, and present an EPTAS for this problem.

Cite As Get BibTex

Hugo A. Akitaya, Justin Dallant, Erik D. Demaine, Michael Kaufmann, Linda Kleist, Frederick Stock, Csaba D. Tóth, and Torsten Ueckerdt. The Price of Connectivity Augmentation on Planar Graphs. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.23

Author Details

Hugo A. Akitaya
  • University of Massachusetts Lowell, MA, USA
Justin Dallant
  • Aarhus University, Denmark
Erik D. Demaine
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Michael Kaufmann
  • Wilhelm-Schickard Institut für Informatik, University of Tübingen, Germany
Linda Kleist
  • Department of Informatics, Universität Hamburg, Germany
Frederick Stock
  • University of Massachusetts Lowell, MA, USA
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, MA, USA
Torsten Ueckerdt
  • Karlsruhe Institute of Technology, Germany

Funding

  • A. Akitaya, Hugo: Research supported in part by NSF grant CCF-2348067.
  • Tóth, Csaba D.: Research supported in part by the NSF award DMS-2154347.
  • Ueckerdt, Torsten: Research supported by the German Research Foundation DFG-520723789.

Acknowledgements

The work in this paper originated at the 11th Annual Workshop on Geometry and Graphs held at the Bellairs Research Institute, 8-15 March, 2024.

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