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Keywords:

  • minimum networks;
  • gradient-constrained networks;
  • Steiner trees;
  • locally minimal

Abstract

The authors have proved in a previous article that the Steiner ratio (the minimum ratio of the length of a Steiner minimal tree to the length of a minimal spanning tree) for a gradient-constrained network on three points is equation image if the maximum gradient m of an edge in the network satisfies m ≤ 1. In this article, we continue this study and use the same strategy to show that this result still holds when the maximum value of m satisfies equation image. However the ratio becomes equation image if equation image. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(4), 354–361