The tropical galaxy of a Laman graph

TL;DR

The paper develops a tropical framework to study the realization number $c_2(G)$ of Laman graphs by showing $2\,c_2(G)=\mathrm{Trop}(G)\cdot(-\mathrm{Trop}(G))$ and introducing the tropical galaxy $\Gamma_G$ with galactic pairing $\langle \Sigma_1,\Sigma_2\rangle = \mathrm{Trop}(\Sigma_1)\cdot(-\mathrm{Trop}(\Sigma_2))$. It analyzes the structural properties of this pairing, including subadditivity and arboreal criteria, and connects these to stars of tropicalizations to obtain lower bounds on $c_2(G)$ via tropical stars. The authors formalize graph excisions as fundamental tropical-building blocks, relate excision chains to chains of flats, and show that arboreal leaf-pairs correspond to unit pairings, with a concrete combinatorial-geometric interpretation. Finally, they provide an open-source software package, TropicalGalaxies.jl, enabling construction of Laman graphs, excisions, tropical galaxies, and star visualizations to enable experimentation and exploration of the proposed framework.

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.

Figures (10)

• Figure 1: Three Laman graphs with $9$ edges.
• Figure 2: Henneberg moves to construct the prism graph.
• Figure 3: The Laman graph $G$, its full excisions $\mathbb{F}_1, \mathbb{F}_2, \mathbb{F}_3$, and their tropicalizations.
• Figure 4: All stars of the polyhedral complex $\Sigma$.
• Figure 5: A Laman graph $G$ and its (iterative) excisions.

Theorems & Definitions (58)

• Definition 2.2
• Example 2.3
• Definition 2.4
• Theorem 2.5: Henneberg1911
• Example 2.6
• Definition 2.7
• Definition 2.8
• Example 2.9
• Lemma 2.10
• proof