  Following the induced gravitational collapse (IGC) paradigm of gamma-ray
bursts (GRBs) associated with type Ib/c supernovae, we present numerical
simulations of the explosion of a carbon-oxygen (CO) core in a binary system
with a neutron-star (NS) companion. The supernova ejecta trigger a
\emph{hypercritical} accretion process onto the NS thanks to a copious neutrino
emission and the trapping of photons within the accretion flow. We show that
temperatures 1--10~MeV develop near the NS surface, hence electron-positron
annihilation into neutrinos becomes the main cooling channel leading to
accretion rates $10^{-9}$--$10^{-1}~M_\odot$~s$^{-1}$ and neutrino luminosities
$10^{43}$--$10^{52}$~erg~s$^{-1}$ (the shorter the orbital period the higher
the accretion rate). We estimate the maximum orbital period, $P_{\rm max}$, as
a function of the NS initial mass, up to which the NS companion can reach by
hypercritical accretion the critical mass for gravitational collapse leading to
black-hole (BH) formation. We then estimate the effects of the accreting and
orbiting NS companion onto a novel geometry of the supernova ejecta density
profile. We present the results of a $1.4\times 10^7$~particle simulation which
show that the NS induces accentuated asymmetries in the ejecta density around
the orbital plane. We elaborate on the observables associated with the above
features of the IGC process. We apply this framework to specific GRBs: we find
that X-ray flashes (XRFs) and binary-driven hypernovae (BdHNe) are produced in
binaries with $P>P_{\rm max}$ and $P < P_{\rm max}$, respectively. We analyze
in detail the case of XRF 060218.
