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Explorations on the number of realizations of minimally rigid graphs

Georg Grasegger

TL;DR

This work investigates how many realizations a minimally rigid graph can have across dimensions by counting complex realizations and studying the effect of rigidity-preserving constructions. It develops and exploits generalized fan constructions to obtain new lower bounds on the maximal realization counts $\mathbf{M}_{d}(n)$ for the plane, sphere, and space, and derives upper bounds for minimal counts in higher dimensions, while leveraging large-scale computations and certificate graphs to validate the results. The authors provide concrete growth-rate bounds, including $\mathbf{M}_{2}(n) \ge 8\cdot2^{(n-5)\mathrm{mod}\,19}\cdot(611930960/8)^{\lfloor(n-5)/19\rfloor}$ (plane) and $\mathbf{M}^\circ_{2}(n) \ge 8\cdot2^{(n-5)\mathrm{mod}\,12}\cdot(1376256/8)^{\lfloor(n-5)/12\rfloor}$ (sphere), as well as a plane-sphere comparison and 3D bounds $\mathbf{M}_{3}(n) \ge 2^{(n-3)\mathrm{mod}\,9}\cdot(54272)^{\lfloor(n-3)/9\rfloor}$. The results rely on probabilistic Gröbner-basis computations for higher dimensions and culminate in a catalogue of certificate graphs and detailed observations about how extension and splitting steps influence realization counts. Overall, the paper advances our understanding of the asymptotics and variability of realization counts and highlights open questions in higher-dimensional rigidity and counting.

Abstract

Rigid graphs have only finitely many realizations. In the recent years significant progress was made in computing the number of such realizations. With this progress it was also possible for the first time to do computations on large sets of graphs. In this paper we show what we can conclude from the data we got from these computations. This includes new lower bounds on the maximal realization count for a given number of vertices, upper bounds for the minimal realization count in higher dimensions and effects of rigidity preserving construction rules on the realization number. In all cases we give certificate graphs which prove the respective results.

Explorations on the number of realizations of minimally rigid graphs

TL;DR

This work investigates how many realizations a minimally rigid graph can have across dimensions by counting complex realizations and studying the effect of rigidity-preserving constructions. It develops and exploits generalized fan constructions to obtain new lower bounds on the maximal realization counts for the plane, sphere, and space, and derives upper bounds for minimal counts in higher dimensions, while leveraging large-scale computations and certificate graphs to validate the results. The authors provide concrete growth-rate bounds, including (plane) and (sphere), as well as a plane-sphere comparison and 3D bounds . The results rely on probabilistic Gröbner-basis computations for higher dimensions and culminate in a catalogue of certificate graphs and detailed observations about how extension and splitting steps influence realization counts. Overall, the paper advances our understanding of the asymptotics and variability of realization counts and highlights open questions in higher-dimensional rigidity and counting.

Abstract

Rigid graphs have only finitely many realizations. In the recent years significant progress was made in computing the number of such realizations. With this progress it was also possible for the first time to do computations on large sets of graphs. In this paper we show what we can conclude from the data we got from these computations. This includes new lower bounds on the maximal realization count for a given number of vertices, upper bounds for the minimal realization count in higher dimensions and effects of rigidity preserving construction rules on the realization number. In all cases we give certificate graphs which prove the respective results.

Paper Structure

This paper contains 23 sections, 27 theorems, 19 equations, 16 figures, 41 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fan constructions
  4. Realizations in the plane
  5. Realizations on the sphere
  6. Comparison of plane and sphere
  7. Realizations in space
  8. Higher dimensions
  9. Changing factors of construction steps
  10. Plane
  11. Extension Constructions
  12. Splittings
  13. Sphere
  14. Extension constructions
  15. Splittings
  16. ...and 8 more sections

Key Result

Lemma 3

Let $G$ be a minimally $d$-rigid graph and $H$ a minimally $d$-rigid proper subgraph with at least $d$ vertices. Then $\mathbf{c}_{d}(H)$ divides $\mathbf{c}_{d}(G)$.

Figures (16)

  • Figure 1: All realizations up to direct isometries of a 4-vertex graph with given edge lengths.
  • Figure 2: Fan construction with four copies of the three-prism graph glued on a triangle subgraph.
  • Figure 3: Growth rates of the lower bounds. The light colors indicate values that were not found by exhaustive search and which therefore could possibly be improved. The horizontal dashed line indicates the lower bound known from LowerBounds. The second vertical dashed line indicates the number of vertices considered in LowerBounds with the respective bound found therein (horizontal dashed line).
  • Figure 4: Graphs with maximal realization count on the sphere $\mathcal{H}^\circ_{2}(n)$ with $n\in\{10,11,12,13\}$ vertices (see \ref{['tab:enc-maxsphere']} for encodings).
  • Figure 5: Growth rates of the lower bounds for realizations on the sphere. The light colors indicate values that were not found by exhaustive search and which therefore could possibly be improved.
  • ...and 11 more figures

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • Theorem 8
  • ...and 29 more