Given a set of sources and a set of sinks as points in the Euclidean plane, a directed network is a directed graph drawn in the plane with a directed path from each source to each sink. Such a network may contain nodes other than the given sources and sinks, called Steiner points. We characterize the local structure of the Steiner points in all shortest-length directed networks in the Euclidean plane. This characterization implies that these networks are constructible by straightedge and compass. Our results build on unpublished work of Alfaro, Campbell, Sher, and Soto from 1989 and 1990. Part of the proof is based on a new method that uses other norms in the plane. This approach gives more conceptual proofs of some of their results, and as a consequence, we also obtain results on shortest directed networks for these norms.

给定一组源和一组汇作为欧几里得平面中的点，有向网络是在平面中绘制的有向图，其中有从每个源到每个接收器的有向路径。这样的网络可能包含除给定源和接收点以外的称为Steiner点的节点。我们表征了欧氏平面上所有最短长度的有向网络中斯坦纳点的局部结构。这种特征意味着这些网络可以用直尺和罗盘构造。我们的结果建立在1989年和1990年Alfaro，Campbell，Sher和Soto尚未发表的工作的基础上。部分证据是基于在飞机上使用其他规范的新方法得出的。这种方法为它们的某些结果提供了更多的概念证明，因此，我们也从这些规范的最短定向网络中获得了结果。