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Discrete & Computational Geometry

, Volume 54, Issue 2, pp 459–480 | Cite as

Compatible Connectivity Augmentation of Planar Disconnected Graphs

  • Greg Aloupis
  • Luis Barba
  • Paz Carmi
  • Vida Dujmović
  • Fabrizio Frati
  • Pat MorinEmail author
Article
  • 106 Downloads

Abstract

We consider the following compatible connectivity-augmentation problem: We are given a labeled n-vertex planar graph \(\mathcal {G}\) that has \(r\ge 2\) connected components, and \(k\ge 2\) isomorphic plane straight-line drawings \(G_1,\ldots ,G_k\) of \(\mathcal {G}\). We wish to augment \(\mathcal {G}\) by adding vertices and edges to make it connected in such a way that these vertices and edges can be added to \(G_1,\ldots ,G_k\) as points and straight line segments, respectively, to obtain k plane straight-line drawings isomorphic to the augmentation of \(\mathcal {G}\). We show that adding \(\varTheta (nr^{1-1/k})\) edges and vertices to \(\mathcal {G}\) is always sufficient and sometimes necessary to achieve this goal. The upper bound holds for all \(r\in \{2,\ldots ,n\}\) and \(k\ge 2\) and is achievable by an algorithm whose running time is \(O(nr^{1-1/k})\) for \(k=O(1)\) and whose running time is \(O(kn^2)\) for general values of k. The lower bound holds for all \(r\in \{2,\ldots ,n/4\}\) and \(k\ge 2\).

Keywords

Planar graphs Graph drawing Connectivity Euclidean minimum spanning trees 

Notes

Acknowledgments

This work was initiated at the Second Workshop on Geometry and Graphs, held at the Bellairs Research Institute, March 9–14, 2014. We are grateful to the other workshop participants for providing a stimulating research environment. We are particularly grateful to Diane Souvaine for bringing references [4] and [5] to our attention. This research was partly supported by NSERC and the Ontario Ministry of Research and Innovation.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Greg Aloupis
    • 1
  • Luis Barba
    • 2
    • 3
  • Paz Carmi
    • 4
  • Vida Dujmović
    • 5
  • Fabrizio Frati
    • 6
  • Pat Morin
    • 2
    Email author
  1. 1.Department of Computer ScienceTufts UniversityMedfordUSA
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  4. 4.Department of Computer ScienceBen-Gurion University of the NegevBeershevaIsrael
  5. 5.School of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada
  6. 6.Department of EngineeringRoma Tre UniversityRomeItaly

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