A tropical approach to rigidity: counting realisations of frameworks

TL;DR

This work addresses counting realisations of graphs as 2D bar-joint frameworks by studying the generic fibre count of the complexified constraint system. It introduces a tropical-geometric formulation in which the 2-realisation number $c_2(G)$ of a minimally 2-rigid graph $G$ is given by a tropical intersection product $2c_2(G)=(-\mathrm{Trop}(M_G))\cdot \mathrm{Trop}(M_G)\cdot \mathrm{Trop}(y_{e}-1)$, where $M_G$ is the graphic matroid of $G$ and $e$ is a fixed edge; this reduces the problem to Bergman fans and stable intersections, enabling bounds via matroid invariants. The authors derive a combinatorial upper bound $2c_2(G)\le \mathrm{nbc}(M_G)$, linking the realisation number to the Tutte polynomial via $\mathrm{nbc}(M_G)=T(M_G;1,0)$, and provide a lower-bound framework through realisation bases, with case studies on graphs like $K_4^-$, the 3-prism, and $K_{3,3}$. Computations on thousands of minimally 2-rigid graphs show the $nbc$-based bound often outperforms mixed-volume bounds, suggesting practical computational advantages and a path toward deeper matroid-theoretic characterisations of realisation numbers. Overall, the paper integrates rigidity, tropical geometry, and matroid theory to illuminate the structure of realisation counts and their invariants, offering both theoretical insights and computational tools for rigidity counting in the plane.

Abstract

A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely-many realisations can be seen as a solution to a system of quadratic equations prescribing the distances between pairs of points. For generic realisations, the size of the solution set depends only on the underlying graph so long as we allow for complex solutions. We provide a characterisation of the realisation number - that is the cardinality of this complex solution set - of a minimally rigid graph. Our characterisation uses tropical geometry to express the realisation number as an intersection of Bergman fans of the graphic matroid. As a consequence, we derive a combinatorial upper bound on the realisation number involving the Tutte polynomial. Moreover, we provide computational evidence that our upper bound is usually an improvement on the mixed volume bound.

Figures (10)

• Figure 1: $K_4^-$ is a minimally 2-rigid graph with 2-realisation number 2.
• Figure 2: On the left and the right are two realisations of the same graph in the plane with different edge lengths. Both sets of edge lengths can be chosen generically. On the left the realisation allows the equivalent realisation in the middle, on the right such a realisation is not possible by the triangle inequality, illustrating that the real realisation number depends on the specific choice of generic realisation.
• Figure 3: A minimally 2-rigid graph $G$ with $c_2(G_2) = 45$. As the number of real equivalent realisations to a generic realisation in $\mathbb{R}^2$ is always even (a corollary of a classical result of Hendrickson hend92), the upper bound on this number given by $c_2(G)$ is not tight.
• Figure 4: $\mathcal{N}(f)$ and $\mathop{\mathrm{Trop}}\nolimits(f)$ for the Laurent polynomial $f$ given in \ref{['ex:tropicalHypersurface']}.
• Figure 5: $\mathop{\mathrm{Trop}}\nolimits(I)$ for $I$ in \ref{['ex:tropicalLinearSpace']}. We exploit the fact that $\mathop{\mathrm{Trop}}\nolimits(I)$ is invariant under translation by $\mathbb{R}\cdot (1,1,1,1)$ to produce pictures in $\mathbb{R}^3$.

Theorems & Definitions (77)

• Proposition 2.1: see, for example, dewar2024number
• Proposition 2.2
• Lemma 2.3: Whiteley:1996
• Theorem 2.4: Pollaczek-Geiringer PollaczekGeiringer
• Proposition 2.5: see, for example, dewar2024number
• Lemma 2.6
• proof
• Example 2.7: Graphic matroids
• Example 2.8
• Example 2.9