Budgeted Steiner Networks: Three Terminals with Equal Path Weights

TL;DR

This paper explores the spectrum of network structures connecting three terminals with equal path weights, bridging the gap between minimal Steiner trees and complete graphs, revealing intermediate configurations.

Contribution

It characterizes the full evolutionary pathway between Steiner trees and complete graphs for three terminals with equal pairwise weights, identifying intermediate structures.

Findings

Identified the full spectrum of network structures between Steiner trees and complete graphs.

Characterized the intermediate configurations for three terminals with equal weights.

Revealed intriguing structural transitions in network connectivity.

Abstract

Given a set of terminals in 2D/3D, the network with the shortest total length that connects all terminals is a Steiner tree. On the other hand, with enough budget, every terminal can be connected to every other terminals via a straight edge, yielding a complete graph over all terminals. In this work, we study a generalization of Steiner trees asking what happens in between these two extremes. Focusing on three terminals with equal pairwise path weights, we characterize the full evolutionary pathway between the Steiner tree and the complete graph, which contains intriguing intermediate structures.