# Compatible Connectivity Augmentation of Planar Disconnected Graphs

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## 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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