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 $\R^d$ 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).
Bibtex entry:
AUTHOR | = | {Dantas Zeus, Guilherme and Jord{\'a}n, Tibor and Silverman, Corwin}, |
TITLE | = | {On generic universal rigidity on the line}, |
NOTE | = | {{\tt egres.elte.hu}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2023}, |
NUMBER | = | {TR-2023-06} |