  We present smoothed-particle-hydrodynamics (SPH) simulations of the
binary-driven hypernova (BdHN) scenario of long gamma-ray bursts (GRBs),
focusing on the binary stability during the supernova (SN) explosion. The BdHN
progenitor is a binary comprised of a carbon-oxygen (CO) star and a neutron
star (NS) companion. The core collapse of the CO leads to an SN explosion and a
newborn NS ($\nu$NS) at its center. Ejected material accretes onto the NS and
the $\nu$NS. BdHNe of type I have compact orbits of a few minutes, the NS
reaches the critical mass, forming a black hole (BH), and the energy release is
$\gtrsim 10^{52}$ erg. BdHNe II have longer periods of tens of minutes to
hours; the NS becomes more massive, remains stable, and the system releases
$\sim 10^{50}$-$10^{52}$ erg. BdHN III have longer periods, even days, where
the accretion is negligible, and the energy released is $\lesssim 10^{50}$ erg.
We assess whether the system remains gravitationally bound after the SN
explosion, leading to an NS-BH in BdHN I, an NS-NS in BdHN II and III, or if
the SN explosion disrupts the system. The existence of bound systems predicts
an evolutionary connection between the long and short GRB populations. We
determine the binary parameters for which the binary remains bound after the
BdHN event. For these binaries, we derive fitting formulas of the numerical
results for the main parameters, e.g., the mass loss, the SN explosion energy,
orbital period, eccentricity, center-of-mass velocity, and the relation between
the initial and final binary parameters, which are useful for outlined
astrophysical applications.
