The $d$-dimensional realisation number of a rigid graph

Abstract

Determining the number of (complex) realisations of a rigid graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we provide two new tools for determining realisation numbers in arbitrary dimensions: (i) we prove that subgraph inclusion translates to realisation number divisibility; and (ii) we provide lower bounds on realisation numbers under specific graph operations in all dimensions. We use these methods to prove that every triangulated sphere with $n$ vertices has at least $2^{n-4}$ edge-length equivalent realisations in 3-dimensions, extending a 2-dimensional result of Jackson and Owen in the case of planar graphs. Additionally, our tools solve a family of conjectures set by Grasegger regarding how 1-extensions, X-replacements, and V-replacements affect realisation numbers.

Figures (7)

• Figure 1: Two different choices of edge lengths for the same graph. The left realisation gives $r(G,p) = 4$: the two realisations pictured plus two more via reflecting the degree 2 vertex through the line passing through its neighbours. The right realisation only gives $r(G,p)=2$: again, the additional realisation can be found via reflecting the degree 2 vertex.
• Figure 2: Three 2-rigid graphs: (left) $G_1$ with $c_2(G_1) = 1$; (middle) $G_2$ with $c_2(G_2) = 45$; (right) $G_3$ with $c_2(G_3) = 32$.
• Figure 3: A schematic of 3-dimensional 0-extension.
• Figure 4: A schematic of the 3-dimensional vertex split.
• Figure 5: A schematic of the 3-dimensional spider split.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2: Borel1991
• Definition 2.3
• Remark 2.4
• Lemma 2.5: dewar23
• Lemma 2.6
• proof