Compatible Connectivity Augmentation of Planar Disconnected Graphs

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