The Transition Region between Brightest Cluster Galaxies and
  Intra-Cluster Light in Galaxy Groups and Clusters
  We take advantage of a state-of-art semi analytic model of galaxy formation,
and the model presented in \citet{contini21a}, to investigate the mass
distribution of Brightest Cluster Galaxies (BCGs) and Intra-Cluster Light (ICL)
by addressing two points: (1) the region of transition between a BCG dominated
distribution and an ICL dominated one, and; (2) the relation between the total
BCG+ICL mass and the ICL one alone. We find the transition radius to be
independent of both BCG+ICL and halo masses, with an average of 60$\pm$40 kpc,
in good agreement with previous observational measurements, but given the large
scatter, it can be considered as a sort of physical separation between the two
components only on cluster scale. From the analysis of $M_{ICL}-M_{BCG+ICL}$
relation, we build a method able to extract the ICL mass directly from the
knowledge of the BCG+ICL one. Given the large scatter on low mass systems, such
method under/overpredicts the true value of the ICL in a significant way, up to
a factor of three in the worst cases. On the other hand, for $\log
M_{BCG+ICL}>12$ or $\log M_{Halo}>14$, the difference between the true value
and the one extracted from the $M_{ICL}-M_{BCG+ICL}$ relation ranges between
$\pm$30\%. We therefore suggest this relation as a reliable test for
observational works aiming to isolate the ICL from the BCG, for systems hosted
by haloes on cluster scale.
