The bottleneck 2-connected kk-Steiner network problem for k≤2k≤2

The geometric bottleneck Steiner network problem on a set of vertices XX embedded in a normed plane requires one to construct a graph GG spanning XX and a variable set of k≥0k≥0 additional points, such that the length of the longest edge is minimised. If no other constraints are placed on GG, then a solution always exists which is a tree. In this paper, we consider the Euclidean bottleneck Steiner network problem for k≤2k≤2, where GG is constrained to be 22-connected. By taking advantage of relative neighbourhood graphs, Voronoi diagrams, and the tree structure of block cut-vertex decompositions of graphs, we produce exact algorithms of complexity O(n2)O(n2) and O(n2logn)O(n2logn) for the cases k=1k=1 and k=2k=2 respectively. Our algorithms can also be extended to other norms such as the LpLp planes.