The number of equivalent realisations of a rigid graph

Given a rigid realisation of a graph $G$ in ${\mathbb R}^2$, 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 ${\mathbb C}^2$ rather that ${\mathbb R}^2$. 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-trivial $3$-edge-cuts. We consider the effect of the Henneberg moves and the vertex-splitting operation on $c(G)$. We use our results to determine $c(G)$ exactly for two important families of graphs, and show that the graphs in both families have $c(G)$ pairwise equivalent generic real realisations. We also show that every planar isostatic graph on $n$ vertices has at least $2^{n-3}$ pairwise equivalent real realisations.