  Configurations of rigid collections of saddle connections are connected
component invariants for strata of the moduli space of quadratic differentials.
They have been classified for strata of Abelian differentials by Eskin, Masur
and Zorich. Similar work for strata of quadratic differentials has been done in
Masur and Zorich, although in that case the connected components were not
distinguished. We classify the configurations for quadratic differentials on
the Riemann sphere and on hyperelliptic connected components of the moduli
space of quadratic differentials. We show that, in genera greater than five,
any configuration that appears in the hyperelliptic connected component of a
stratum also appears in the non-hyperelliptic one.
