Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

Connectivity augmentation is a classical problem in combinatorial optimization (see [4, 5]). Given a graph G = (V,E) and a parameter τ ∈ N, add a set of new edges E+ such that the augmented graph G′ = (V,E ∪ E+) is τ -connected (resp., τ -edge-connected). Over planar straightline graphs (PSLGs), it is NP-complete to find the minimum number of edges for τ -connectivity or τ -edge-connectivity augmentation for 2 ≤ τ ≤ 5 [7]. In the worst case, b(2n − 2)/3c (resp., b(4n − 4)/3c) new edges can augment a connected (resp., disconnected) PSLG to 2-edgeconnectivity [2, 8]. A tight bound of b − 1 is known for augmenting a connected PSLG G to 2-connectivity [1], where b is the number of 2connected components in G. Motivated by communication networks, Dobrev et al. [3] studied the question of producing a 2-edge-connected PSLG that minimizes the length of the longest edge and Kranakis et al. [6] studied the combined problem of adding the minimum number of straight-line edges of bounded length. In this paper we consider weighted connectivity augmentation for PSLGs. We prove the following theorems, where ‖E‖ denotes the sum of edge lengths in E and MST(V ) denotes the Euclidean minimum spanning tree of V .