关于最短有向网络的存在性
On the existence of shortest directed networks
摘要：
一个连接集合A和集合B的有向网络是一个有向图，它包含每个a在A中和b在B中的a-b路。有向网络中不在A或B中的顶点称为Steiner点。本文证明了在一个有测地线存在的紧度量空间中，任意两个有限集A和B都由一个最短有向网连接。我们还通过A和B的大小的函数来限制Steiner点的数量。以前，这样的存在性结果只知道欧几里德平面[M。Alfaro，Pacific J.Math.167（1995）201-214]。主要的困难是，与无向情况（Steiner最小树）不同，底层图不需要是无环的。在无向情况下的存在性首先由E. J. Cockayne [Canad.Math.Bull.10（1967）431-450].
英文摘要：
A directed network connecting a set A to a set B is a digraph containing an a-b path for each a in A and b in B. Vertices in the directed network not in A or B are called Steiner points. We show that in a finitely compact metric space in which geodesics exist, any two finite sets A and B are connected by a shortest directed network. We also bound the number of Steiner points by a function of the sizes of A and B. Previously, such an existence result was known only for the Euclidean plane [M. Alfaro, Pacific J. Math. 167 (1995) 201-214]. The main difficulty is that, unlike the undirected case (Steiner minimal trees), the underlying graphs need not be acyclic. Existence in the undirected case was first shown by E. J. Cockayne [Canad. Math. Bull. 10 (1967) 431-450].
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