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 $\tilde{\mathcal{P}}_{\min}$.