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.

Additional Metadata
Keywords Connectivity, Euclidean minimum spanning trees, Graph drawing, Planar graphs
Persistent URL dx.doi.org/10.1007/s00454-015-9716-8
Journal Discrete and Computational Geometry
Aloupis, G, Barba, L. (Luis), Carmi, P. (Paz), Dujmović, V, Frati, F. (Fabrizio), & Morin, P. (2015). Compatible Connectivity Augmentation of Planar Disconnected Graphs. Discrete and Computational Geometry, 54(2), 459–480. doi:10.1007/s00454-015-9716-8