The Steiner minimal network for convex configurations.(English)Zbl 0774.05033

If $$X$$ is a finite set of points in the Euclidean plane, the Steiner problem is to find a shortest network connecting the points. In general, this problem is NP-hard. In a former paper by D. A. Thomas and J. H. Rubinstein [ibid. 7, No. 1, 77-86 (1992; see the review above)] it is shown that the Steiner problem is much easier, if $$X$$ lies on a circle. Generalizing this result the paper shows: Suppose $$X$$ is a convex configuration with radius of maximum curvature $$r$$ and at most one of the edges joining neighboring points has length strictly greater than $$r$$, then the shortest network (the Steiner tree) consists of all the edges with a longest edge removed.

MSC:

 05C05 Trees 52A37 Other problems of combinatorial convexity

Zbl 0774.05031
Full Text:

References:

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