The bottleneck 2-connected $k$-Steiner network problem for $k\le 2$

Abstract

The geometric bottleneck Steiner network problem on a set of vertices $X$ embedded in a normed plane requires one to construct a graph $G$ spanning $X$ and a variable set of $k\ge 0$ additional points, such that the length of the longest edge is minimised. If no other constraints are placed on $G$, then a solution always exists which is a tree. In this paper, we consider the Euclidean bottleneck Steiner network problem for $k\le 2$, where $G$ is constrained to be $2$-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 $O\left(n^2\right)$ and $O(n^{2}logn)$ for the cases $k=1$ and $k=2$ respectively. Our algorithms can also be extended to other norms such as the $L_p$ planes.