# Shortest directed networks in the plane

## TL;DR

This paper characterizes the local structure of Steiner points in shortest directed networks in the Euclidean plane, showing they are constructible with straightedge and compass and extending results to other norms.

## Contribution

It provides a complete characterization of Steiner points in shortest directed networks and introduces a new method using different norms for analysis.

## Key findings

- Steiner points have a specific local structure in shortest networks.
- Networks are constructible by straightedge and compass.
- Results extend to networks under different norms.

## Abstract

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 Our results build on unpublished work of Alfaro, Campbell, Sher, and Soto from 1989 and 1990.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.