  In the Edwards-Anderson model of spin glasses with a bimodal distribution of
bonds, the degeneracy of the ground state allows one to define a structure
called backbone, which can be characterized by the rigid lattice (RL),
consisting of the bonds that retain their frustration (or lack of it) in all
ground states. In this work we have performed a detailed numerical study of the
properties of the RL, both in two-dimensional (2D) and three-dimensional (3D)
lattices. Whereas in 3D we find strong evidence for percolation in the
thermodynamic limit, in 2D our results indicate that the most probable scenario
is that the RL does not percolate. On the other hand, both in 2D and 3D we find
that frustration is very unevenly distributed. Frustration is much lower in the
RL than in its complement. Using equilibrium simulations we observe that this
property can be found even above the critical temperature. This leads us to
propose that the RL should share many properties of ferromagnetic models, an
idea that recently has also been proposed in other contexts. We also suggest a
preliminary generalization of the definition of backbone for systems with
continuous distributions of bonds, and we argue that the study of this
structure could be useful for a better understanding of the low temperature
phase of those frustrated models.
