A necessary condition for generic rigidity of bar-and-joint frameworks in $d$-space

Details A necessary condition for generic rigidity of bar-and-joint frameworks in $d$-space A necessary condition for generic rigidity of bar-and-joint frameworks in $d$-space

2011-04-22

arxiv.org/abs/1104.4415v2

Mathematics

Combinatorics

There was an error in the proof of Theorem 3.3(b) in version 1 of this paper. I have replaced Lemma 3.2 in version 1 by Lemmas

Scientific paper

A graph $G=(V,E)$ is {$d$-sparse} if each subset $X\subseteq V$ with $|X|\geq d$ induces at most $d|X|-{{d+1}\choose{2}}$ edges in $G$. Laman showed in 1970 that a necessary and sufficient condition for a realisation of $G$ as a generic bar-and-joint framework in $\real^2$ to be rigid is that $G$ should have a 2-sparse subgraph with $2|V|-3$ edges. Although Laman's theorem does not hold when $d\geq 3$, Cheng and Sitharam recently showed that if $G$ is generically rigid in $\real^3$ then every maximal 3-sparse subgraph of $G$ must have $3|V|-6$ edges. We extend their result to all $d\leq 5$ by showing that if $G$ is generically rigid in $\real^d$ then every maximal $d$-sparse subgraph of $G$ must have $d|V|-{{d+1}\choose{2}}$ edges.

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