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Regular Augmentation of Planar Graphs

Algorithmica volume 73, pages 306–370 (2015)Cite this article

Abstract

In this paper, we study the problem of augmenting a planar graph such that it becomes \(k\)-regular, \(c\)-connected and remains planar, either in the sense that the augmented graph is planar, or in the sense that the input graph has a fixed (topological) planar embedding that can be extended to a planar embedding of the augmented graph. We fully classify the complexity of this problem for all values of \(k\) and \(c\) in both, the variable embedding and the fixed embedding case. For \(k \le 2\) all problems are simple and for \(k \ge 4\) all problems are NP-complete. Our main results are efficient algorithms (with running time \(O(n^{1.5}))\) for deciding the existence of a \(c\)-connected, 3-regular augmentation of a graph with a fixed planar embedding for \(c=0,1,2\) and a corresponding hardness result for \(c=3\). The algorithms are such that for yes-instances a corresponding augmentation can be constructed in the same running time.

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Acknowledgments

We thank an anonymous referee for several suggestions that helped us to simplify and shorten the algorithm in Sect. 2 and the connectivity proofs in Sect. 7.

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Authors and Affiliations

  1. Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Tanja Hartmann, Jonathan Rollin & Ignaz Rutter

Authors
  1. Tanja Hartmann
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  2. Jonathan Rollin
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  3. Ignaz Rutter
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Corresponding author

Correspondence to Ignaz Rutter.

Additional information

Partially supported by the DFG under grant WA 654/15 within the Priority Programme “Algorithm Engineering”.

A preliminary version of this paper has appeared as T. Hartmann, J. Rollin, I. Rutter, Cubic Augmentation of Planar Graphs, In Proceedings of the 23rd International Symposium on Algorithms and Computation (ISAAC’12), pages 402–412, volume 7676 of LNCS, 2013.

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Hartmann, T., Rollin, J. & Rutter, I. Regular Augmentation of Planar Graphs. Algorithmica 73, 306–370 (2015). https://doi.org/10.1007/s00453-014-9922-4

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  • DOI: https://doi.org/10.1007/s00453-014-9922-4

Keywords

  • Regular planar graphs
  • Graph augmentation
  • Complexity
  • Efficient algorithms
  • Connectivity