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In this paper we study the Steiner minimal tree T problem for a point set Z with cardinality n and one polygonal obstacle ω in the Euclidean plane. We assume ω touches only one convex path in T that joins two terminals and that the number of extreme points of the obstacle is k . If all degree 2 vertices are omitted, then the topology of T is called the(More)
Fifty years ago Jarnik and K6ssler showed that a Steiner minimal tree for the vertices of a regular n-gon contains Steiner points for 3 <n <5 and contains no Steiner point for n = 6 and n -> 13. We complete the story by showing that the case for 7 -< n -< 12 is the same as n -> 13. We also show that the set of n equally spaced points yields the longest(More)
The Gilbert network problem is a generalization of the Steiner minimal tree problem derived by adding flow-dependent weights to the edges. In this paper, we define a special class of minimum Gilbert networks, called pseudo-Gilbert–Steiner trees, and we show that it can be constructed by Gilbert’s generalization of Melzak’s method. Besides, a counterexample,(More)
A Steiner tree T on a given set of points A is called linear if all Steiner points, including those collapsing into their adjacent given points, lie on one path referred to as its trunk. Suppose A is a simple polygonal line. Roughly speaking, T is similar to A if its trunk turns right or left when A does. In this paper we prove that A can be expanded to(More)