# Steiner Trees, Coordinate Systems and NP-Hardness

• J. F. Weng
Part of the Combinatorial Optimization book series (COOP, volume 6)

## Abstract

Given a set A of points a 1, a 2,..., in the Euclidean plane, the Steiner tree problem asks for a minimum network T(A) (or T if A is not necessarily mentioned) interconnecting A with some additional points to shorten the network [6]. The given points are referred to as terminals and the additional points are referred to as Steiner points. Trivially, T is a tree, called the Euclidean Steiner minimal tree (ESMT) for A. It is well known that Steiner minimal trees satisfy an angle condition: all angles at the vertices of Steiner minimal trees are not less than 120° [6]. A tree satisfying this angle condition is called a Steiner tree. Therefore, a Steiner minimal tree must be a Steiner tree.

Hexagonal

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