Regular Augmentation of Planar Graphs

In this paper, we study the problem of augmenting a planar graph such that it becomes $$k$$k-regular, $$c$$c-connected and remains planar, either in the sense that the augmented graph is planar, or in the sense that the input graph has a fixed (topological) planar embedding that can be extended to a planar embedding of the augmented graph. We fully classify the complexity of this problem for all values of $$k$$k and $$c$$c in both, the variable embedding and the fixed embedding case. For $$k \le 2$$k≤2 all problems are simple and for $$k \ge 4$$k≥4 all problems are NP-complete. Our main results are efficient algorithms (with running time $$O(n^{1.5}))$$O(n1.5)) for deciding the existence of a $$c$$c-connected, 3-regular augmentation of a graph with a fixed planar embedding for $$c=0,1,2$$c=0,1,2 and a corresponding hardness result for $$c=3$$c=3. The algorithms are such that for yes-instances a corresponding augmentation can be constructed in the same running time.