Compatible connectivity-augmentation of planar disconnected graphs

Motivated by applications to graph morphing, we consider the following compatible connectivity-augmentation problem: We are given a labelled n-vertex planar graph, Q, that has τ ≥ 2 connected components, and k ≥ 2 isomorphic planar straight-line drawings, G₁,⋯ ,G_k, of Q. We wish to augment Q by adding vertices and edges to make it connected in such a way that these vertices and edges can be added to G₁⋯G_k as points and straight-line segments, respectively, to obtain k planar straight-line drawings isomorphic to the augmentation of Q. We show that adding θ (nr^1-1/k) edges and vertices to Q is always sufficient and sometimes necessary to achieve this goal. The upper bound holds for all τ ∈ {2⋯ n} and k ≥ 2 and is achievable by an algorithm whose running time is θ (nr^1-1/k) for k = O (l) and whose running time is 0 (kn²) for general values of k. The lower bound holds for all τ ∈ {2,⋯, n/4} and k ≥ 2.