The tropical galaxy of a Laman graph

TL;DR

This paper introduces the concept of the tropical galaxy of a Laman graph to analyze its realization number, providing new structural insights and a software tool for tropical geometric computations.

Contribution

It defines the tropical galaxy of a Laman graph, explores its properties, and links it to the realization number, advancing the understanding of graph rigidity through tropical geometry.

Findings

Tropical galaxy provides lower bounds for realization number.

Galactic pairing relates to arboreal pairs and subadditivity.

Software package enables practical computations with tropical galaxies.

Abstract

A Laman graph $G$ is a minimally rigid graph in dimension two, and its realization number is its number of distinct embeddings with fixed generic edge lengths. While conjectured to grow exponentially in the number of vertices of $G$, the best proven lower bound is merely $2$. Motivated by the fact that the realization number can be expressed as a tropical intersection product involving $\mathrm{Trop}(G)$, the Bergman fan of the graphic matroid of $G$, and the fact that stars of $\mathrm{Trop}(G)$ naturally lead to lower bounds thereof, we introduce the tropical galaxy of $G$ together with a galactic pairing thereon. We study structural properties of this pairing, such as under which conditions it is non-trivially subadditive, and connect it being non-zero to arboreal pairs. We also present a software package for working with tropical galaxies.