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A 2-dimensional framework $(G,p)$ is a graph $G=(V,E)$ together with a map $p:V\to {\mathbb{R}}^2$. We consider the framework
to be a straight line realization of $G$ in $\mathbb{R}^2$.
Two realizations of $G$ are equivalent if the corresponding edges
in the two frameworks have the same length.
A pair of vertices $\{u,v\}$ is globally linked in $G$ if the distance between the points corresponding to $u$ and $v$ is the same
in all pairs of equivalent generic realizations of $G$.
In this paper we extend our previous results on globally linked
pairs and complete the characterization of globally linked pairs in
minimally rigid graphs. We also show that the Henneberg
1-extension operation on a non-redundant edge preserves the
property of being not globally linked, which can be used to identify
globally linked pairs in broader families of graphs. We prove that
if $(G,p)$ is generic then the set of globally linked pairs does not
change if we perturb the coordinates slightly. Finally, we
investigate when we can choose a non-redundant edge $e$ of $G$ and
then continuously deform a generic realization of $G-e$ to obtain
equivalent generic realizations of $G$ in which the distances
between a given pair of vertices are different.
Bibtex entry:
| AUTHOR | = | {Jackson, Bill and Jord{\'a}n, Tibor and Szabadka, Zolt{\'a}n}, |
| TITLE | = | {Globally linked pairs of vertices in rigid frameworks}, |
| NOTE | = | {{\tt egres.elte.hu}}, |
| INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
| YEAR | = | {2012}, |
| NUMBER | = | {TR-2012-19} |