  In this paper, we have examined the interplay between the lobby size $q$ in
the $q$-neighbor Ising model of opinion formation [Phys. Rev. E 92, 052105] and
the level of overlap $v$ of two fully connected graphs. Results suggest that
for each lobby size $q \ge 3$ there exists a specific level of overlap $v^*$
which destroys initially polarized clusters of opinions. By performing
Monte-Carlo simulations, backed by an analytical approach we show that the
dependence of the $v^*$ on the lobby size $q$ is far from trivial in the
absence of temperature $T \rightarrow 0$, showing a clear maximum that
additionally depends on the parity of $q$. On the other hand, the temperature
is a destructive factor, its increase leads to the earlier collapse of
polarized clusters but additionally brings a substantial decrease in the level
of polarization.
