Steiner minimal trees on sets of four points
type: Item
instance of: scholarly article
title: Steiner minimal trees on sets of four points
zbMATH Open document ID: 0623.05013
DOI: 10.1007/BF02187892
author: Du Ding-Zhu
author: G. Y. Ting
author: Frank K. Hwang
author: Guo-Dong Song
published in: Discrete \& Computational Geometry
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131031
review text: Let \(S=\{A,B,C,D\}\) consist of the four corner points of a convex quadrilateral where diagonals [A,C] and [B,D] intersect at the point O. There are two possible full Steiner trees for S, the AB-CD tree has A and B adjacent to one Steiner point, and C and D to another; the AD-BC tree has A and D adjacent to one Steiner point, and B and C to another. Pollak proved that if both full Steiner trees exist, then the AB-CD (AD-BC) tree is the Steiner minimal tree if \(\sphericalangle AOD>(<)90\circ\), and both are Steiner minimal trees if \(\sphericalangle AOD=90\circ\). While the theorem has been crucially used in obtaining results on Steiner minimal trees in general, its applicability is sometimes restricted because of the condition that both full Steiner trees must exist. In this paper we remove this obstacle by showing: (i) Necessary and sufficient conditions for the existence of either full Steiner tree for S. (ii) If \(\sphericalangle AOD\geq 90\circ\), then the AB-CD tree is the SMT even if the AD-BC tree does not exist. (iii) If \(\sphericalangle AOD<90\circ\) but the AD-BC tree does not exist, then the AB-CD tree cannot be ruled out as a Steiner minimal tree, though under certain broad conditions it can.
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 52Bxx
zbMATH DE Number: 4010535
zbMATH Keywords: Steiner trees
zbMATH Keywords: Steiner minimal tree
MaRDI profile type: Publication
cites work: Q3689203
cites work: Steiner Minimal Trees on Zig-Zag Lines
cites work: Steiner minimal trees for bar waves
cites work: The Complexity of Computing Steiner Minimal Trees
cites work: Steiner Minimal Trees
cites work: On the Problem of Steiner
cites work: Some remarks on the Steiner problem
OpenAlex ID: W2090953203
description: scientific article; zbMATH DE number 4010535