Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

Abstract

We consider edge insertion and deletion operations that increase the connectivity of a given planar straight-line graph (PSLG), while minimizing the total edge length of the output. We show that every connected PSLG G = (V, E) in general position can be augmented to a 2connected PSLG (V, E ∪ E + ) by adding new edges of total Euclidean length E + ≤ 2 E , and this bound is the best possible. An optimal edge set E + can be computed in O(|V | 4 ) time; however the problem becomes NP-hard when G is disconnected. Further, there is a sequence of edge insertions and deletions that transforms a connected PSLG G = (V, E) into a planar straight-line cycle G = (V, E ) such that E ≤ 2 MST(V ) , and the graph remains connected with edge length below E + MST(V ) at all stages. These bounds are the best possible. IntroductionConnectivity augmentation is a classical problem in combinatorial optimization. Given a graph G = (V, E) and a parameter τ ∈ N, add a set of new edges E + of minimum cardinality or weight such that the augmented graph G = (V, E ∪ E + ) is τ -connected (resp., τ -edge-connected). Efficient algorithms are known for both connectivity and edge-connectivity augmentation over abstract graphs and constant τ [8,19,24]. In this paper we consider weighted connectivity augmentation for planar straight-line graphs (PSLGs). The vertices are points in Euclidean plane, the edges are noncrossing line segments between the corresponding vertices, and the weight of an edge is its Euclidean length.The edge-and node-connectivity of a planar graph is at most 5 by Euler's theorem. Further, not every PSLG can be augmented to a 3-connected (resp., 4-edge-connected) PSLG; see [12] for feasibility conditions. Finding the minimum number of edges to augment a given PSLG to τconnectivity or τ -edge-connectivity is NP-complete [22] for 2 ≤ τ ≤ 5; the reduction requires the input graph G to be disconnected (the NP-hardness claim for connected input [22, Corollary 2] turned out to be flawed). Worst case bounds are known for the most important cases: Every PSLG G with n vertices can be augmented to 2-edge-connectivity with at most (4n − 4)/3 edges [2];