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 ($ u$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 $ u$NS from SN matter fallback. We determine the fate of the binary system for a wide parameter space including different CO$_{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 $ u$NS-NS and $ u$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.
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