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## Augmenting the Connectivity of Planar and Geometric Graphs

Please always quote using this URN: urn:nbn:de:bvb:20-opus-97587
• In this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments. We show that it is NP-hard to � nd a minimum-cardinality augmentationIn this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments. We show that it is NP-hard to � nd a minimum-cardinality augmentation that makes a planar graph 2-edge connected. For making a planar graph 2-vertex connected this was known. We further show that both problems are hard in the geometric setting, even when restricted to trees. The problems remain hard for higher degrees of connectivity. On the other hand we give polynomial-time algorithms for the special case of convex geometric graphs. We also study the following related problem. Given a planar (plane geometric) graph G, two vertices s and t of G, and an integer c, how many edges have to be added to G such that G is still planar (plane geometric) and contains c edge- (or vertex-) disjoint s{t paths? For the planar case we give a linear-time algorithm for c = 2. For the plane geometric case we give optimal worst-case bounds for c = 2; for c = 3 we characterize the cases that have a solution.  Author: Alexander Wolff, Iganz Rutter urn:nbn:de:bvb:20-opus-97587 Journal article Fakultät für Mathematik und Informatik / Institut für Informatik English Journal of Graph Algorithms and Applications 2012 In: Journal of Graph Algorithms and Applications (2012) 16: 2, 599-628, DOI: 10.755/jgaa.00275 https://doi.org/10.7155/jgaa.00275 0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 004 Datenverarbeitung; Informatik 2014/05/12 Deutsches Urheberrecht