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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 $k\geq 0$ additional points, such that the length of thelongest edge is minimised. If no other constraints are placed on $G$ then asolution always exists which is a tree. In this paper we consider the Euclideanbottleneck Steiner network problem for $k\leq 2$, where $G$ is constrained tobe 2-connected. By taking advantage of relative neighbourhood graphs, Voronoidiagrams, and the tree structure of block cut-vertex decompositions of graphs,we produce exact algorithms of complexity $O(n^2)$ and $O(n^2\log n)$ for thecases $k=1$ and $k=2$ respectively. Our algorithms can also be extended toother norms such as the $L_p$ planes.

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