For a given set of points in the Euclidean plane, a minimum network (a Steiner minimal tree) can be constructed using a geometric method, called Melzak's construction. The core of the Melzak construction is to replace a pair of terminals adjacent to the same Steiner point with a new terminal. In this paper we prove that the Melzak construction can be generalized to constructing Steiner minimal trees for circles so that either the given points (terminals) are constrained on the circles or the terminal edges are tangent to the circles. Then we show that the generalized Melzak construction can be used to find minimum networks separating and surrounding circular objects or to find minimum networks connecting convex and smoothly bounded objects and avoiding convex and smoothly bounded obstacles.
An approximation algorithm for a bottleneck k-Steiner tree problem in the Euclidean plane
