The number of equivalent realisations of a rigid graph

Details The number of equivalent realisations of a rigid graph The number of equivalent realisations of a rigid graph

2012-04-05

arxiv.org/abs/1204.1228v1

Mathematics

Combinatorics

Scientific paper

Given a generic rigid realisation of a graph in $\real^2$, it is an open problem to determine the maximum number of pairwise non-congruent realisations which have the same edge lengths as the given realisation. This problem can be restated as finding the number of solutions of a related system of quadratic equations and in this context it is natural to consider the number of solutions in $\complex^2$. We show that the number of complex solutions, $c(G)$, is the same for all generic realisations of a rigid graph $G$, characterise the graphs $G$ for which $c(G)=1$, and show that the problem of determining $c(G)$ can be reduced to the case when $G$ is 3-connected and has no non-trival 3-edge-cuts. We also consider the effect of the the so called Henneberg moves on $c(G)$ and determine $c(G)$ exactly for two important families of graphs.

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