  We consider a restricted game on weighted graphs associated with minimum
partitions. We replace in the classical definition of Myerson restricted game
the connected components of any subgraph by the sub-components obtained with a
specific partition $\tilde{\mathcal{P}}_{\min}$. This partition relies on the
same principle as the partition $\mathcal{P}_{\min}$ introduced by Grabisch and
Skoda (2012) but restricted to connected coalitions. More precisely, this new
partition $\tilde{\mathcal{P}}_{\min}$ is induced by the deletion of the
minimum weight edges in each connected component associated with a coalition.
We provide a characterization of the graphs satisfying inheritance of convexity
from the underlying game to the restricted game associated with
$\tilde{\mathcal{P}}_{\min}$.
