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.


05C05 Trees
52A37 Other problems of combinatorial convexity


Zbl 0774.05031
Full Text: DOI EuDML


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