## Geometric Sensitivity of Rigid Graphs

#### Tibor Jordán, Gábor Domokos, Krisztina Tóth

Published in:
SIAM J. Discrete Math., vol. 27, no. 4, pp. 1710-1726, 2013.

### Abstract

Let $(G,p)$ be an infinitesimally rigid $d$-dimensional bar-and-joint framework and let $L$ be an equilibrium load on $p$. The load can be resolved by appropriate stresses $w_{i,j}$, $ij\in E(G)$, in the bars of the framework. Our goal is to identify the following parts (zones) of the framework:

(i) when the location of an unloaded joint $v$ is slightly perturbed, and the same load is applied, the stress will change in some of the bars. We call the set of these bars the {\it influenced zone of} $v$ (with respect to $L, p$ and the modified configuration $p'$),

(ii) let $S$ be a designated set of joints and suppose that each joint with a non-zero load belongs to $S$. The {\it active zone} of $S$ (with respect to $p$ and $L$) is the set of those bars in which the stress, which resolves $L$, is non-zero.

We show that if $(G,p)$ is generic and $d=2$ then, for almost all loads, these zones depend only on the graph $G$ of the framework and can be computed by efficient combinatorial methods.

Bibtex entry:

@techreport{egres-11-12,
AUTHOR = {Jord{\'a}n, Tibor and Domokos, G{\'a}bor and T{\'o}th, Krisztina},
TITLE = {Geometric Sensitivity of Rigid Graphs},
NOTE= {{\tt www.cs.elte.hu/egres}},
INSTITUTION = {Egerv{\'a}ry Research Group, Budapest},
YEAR = {2011},
NUMBER = {TR-2011-12}
}