  We study the growth pathways of Brightest Central Galaxies (BCGs) and
Intra-Cluster Light (ICL) by means of a semi-analytic model. We assume that the
ICL forms by stellar stripping of satellite galaxies and violent processes
during mergers, and implement two independent models: (1) one considers both
mergers and stellar stripping (named {\small STANDARD} model), and one
considers only mergers (named {\small MERGERS} model). We find that BCGs and
ICL form, grow and overall evolve at different times and with different
timescales, but they show a clear co-evolution after redshift $z \sim 0.7-0.8$.
Around 90\% of the ICL from stellar stripping is built-up in the innermost 150
Kpc from the halo centre and the dominant contribution comes from disk-like
galaxies (B/T$<$0.4) through a large number of small/intermediate stripping
events ($M_{strip}/M_{sat}<0.3$). The fractions of stellar mass in BCGs and in
ICL over the total stellar mass within the virial radius of the halo evolve
differently with time. At high redshift, the BCG accounts for the bulk of the
mass, but its contribution gradually decreases with time and stays constant
after $z\sim 0.4-0.5$. The ICL, instead, grows very fast and its contribution
keeps increasing down to the present time. The {\small STANDARD} and the
{\small MERGERS} models make very similar predictions in most of the cases, but
predict different amounts of ICL associated to other galaxies within the virial
radius of the group/cluster other than the BCG, at $z=0$. We then suggest that
this quantity is a valid observable that can shed light on the relative
importance of mergers and stellar stripping for the formation of the ICL.
