  We present the first three-dimensional (3D) smoothed-particle-hydrodynamics
(SPH) simulations of the induced gravitational collapse (IGC) scenario of
long-duration gamma-ray bursts (GRBs) associated with supernovae (SNe). We
simulate the SN explosion of a carbon-oxygen core (CO$_{\rm core}$) forming a
binary system with a neutron star (NS) companion. We follow the evolution of
the SN ejecta, including their morphological structure, subjected to the
gravitational field of both the new NS ($\nu$NS) formed at the center of the
SN, and the one of the NS companion. We compute the accretion rate of the SN
ejecta onto the NS companion as well as onto the $\nu$NS from SN matter
fallback. We determine the fate of the binary system for a wide parameter space
including different CO$_{\rm core}$ and NS companion masses, orbital periods
and SN explosion geometry and energies. We identify, for selected NS nuclear
equations-of-state, the binary parameters leading the NS companion, by
hypercritical accretion, either to the mass-shedding limit, or to the secular
axisymmetric instability for gravitational collapse to a black hole (BH), or to
a more massive, fast rotating, stable NS. We also assess whether the binary
remains or not gravitationally bound after the SN explosion, hence exploring
the space of binary and SN explosion parameters leading to $\nu$NS-NS and
$\nu$NS-BH binaries. The consequences of our results for the modeling of long
GRBs, i.e. X-ray flashes and binary-driven hypernovae, are discussed.
