## Abstract

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

Original language | English |
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Pages (from-to) | 459-480 |

Number of pages | 22 |

Journal | Discrete and Computational Geometry |

Volume | 54 |

Issue number | 2 |

DOIs | |

State | Published - 27 Sep 2015 |

## Keywords

- Connectivity
- Euclidean minimum spanning trees
- Graph drawing
- Planar graphs

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics