, Volume 73, Issue 2, pp 306–370 | Cite as

Regular Augmentation of Planar Graphs



In this paper, we study the problem of augmenting a planar graph such that it becomes \(k\)-regular, \(c\)-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 \(k\) and \(c\) in both, the variable embedding and the fixed embedding case. For \(k \le 2\) all problems are simple and for \(k \ge 4\) all problems are NP-complete. Our main results are efficient algorithms (with running time \(O(n^{1.5}))\) for deciding the existence of a \(c\)-connected, 3-regular augmentation of a graph with a fixed planar embedding for \(c=0,1,2\) and a corresponding hardness result for \(c=3\). The algorithms are such that for yes-instances a corresponding augmentation can be constructed in the same running time.


Regular planar graphs Graph augmentation Complexity  Efficient algorithms Connectivity 



We thank an anonymous referee for several suggestions that helped us to simplify and shorten the algorithm in Sect. 2 and the connectivity proofs in Sect. 7.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Karlsruhe Institute of Technology (KIT)KarlsruheGermany

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