On generic universal rigidity on the line

TL;DR

This work analyzes universal rigidity of 1D bar-and-joint frameworks, showing that Connelly's inductive characterization fails by constructing counterexamples and introducing two new graph-operations that control universal rigidity. It proves that the join along $k$ edges preserves generic universal rigidity when $k\ge 4$, and that degree-2 extension preserves universal rigidity under a nondegeneracy condition, enabling infinite families of generically universally rigid graphs and a nontrivial edge bound. A key result is $|E| \ge \frac{3}{2}|V|$ for $|V| \ge 6$, advancing understanding toward Jordán and Nguyen's question. The paper also presents the augmented Grötzsch graph as a triangle-free, generically universally rigid example on the line, illustrating the broader landscape of 1D GUR graphs and suggesting avenues for further exploration in higher dimensions and girth questions.

Abstract

A $d$-dimensional bar-and-joint framework $(G,p)$ with underlying graph $G$ is called universally rigid if all realizations of $G$ with the same edge lengths, in all dimensions, are congruent to $(G,p)$. A graph $G$ is said to be generically universally rigid in $\mathbb{R}^d$ if every $d$-dimensional generic framework $(G,p)$ is universally rigid. In this paper we focus on the case $d=1$. We give counterexamples to a conjectured characterization of generically universally rigid graphs from R. Connelly (2011). We also introduce two new operations that preserve the universal rigidity of generic frameworks, and the property of being not universally rigid, respectively. One of these operations is used in the analysis of one of our examples, while the other operation is applied to obtain a lower bound on the size of generically universally rigid graphs. This bound gives a partial answer to a question from T. Jordán and V-H. Nguyen (2015).

Figures (6)

• Figure 1: Realizations of a four-cycle.
• Figure 2: Graphs $B_3$, $B_4$, and $B_5$.
• Figure 3: The Grötzsch graph and the augmented Grötzsch graph. The central vertex is $w$.
• Figure 4: The graph obtained from $G$ and $H$ by a join operation along $k$ edges.
• Figure 5:

Theorems & Definitions (16)

• Conjecture 1
• Lemma 2
• proof
• Definition 3
• Lemma 4
• Theorem 5
• Theorem 6
• proof
• Theorem 7
• proof