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Adiprasito and Nevo (2018) proved that there exists a set of 76 points
in $\R^3$ such that every triangulated planar graph has an
infinitesimally rigid realization in which each vertex
is mapped to a point in this set.
In this paper we show that there exists a set $A$ of 26 points
in the plane such that every planar graph which is generically rigid in $\R^2$ has an
infinitesimally rigid realization in which each vertex
is mapped to a point in this set.
It is known that a similar result, with
a set of constant size, does not hold for the family of
all generically rigid graphs in $R^d$, $d\geq 2$.
We show that for every positive integer $n$ there is a set of
$c(\sqrt{n})$ points in the plane, for some constant $c$, such that every generically rigid graph
in $R^2$ has an infinitesimally rigid realization on this set.
This bound is best possible.
Bibtex entry:
| AUTHOR | = | {Kir{\'a}ly, Csaba}, |
| TITLE | = | {Rigid realizations of graphs with few locations in the plane}, |
| NOTE | = | {{\tt egres.elte.hu}}, |
| INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
| YEAR | = | {2019}, |
| NUMBER | = | {TR-2019-13} |