Compatible Connectivity Augmentation of Planar Disconnected Graphs

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.