Augmenting the Connectivity of Planar and Geometric Graphs

In this paper we study some 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 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 find 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 also 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) grap...

• Type: Journal
• Year: 2008
• Where: ENDM
• Authors: Ignaz Rutter, Alexander Wolff
• Keywords: ENDM 2008, Geometric, Geometric Setting, Planar Graph