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We consider a restricted game on weighted graphs associated with minimumpartitions. We replace in the classical definition of Myerson restricted gamethe connected components of any subgraph by the sub-components obtained with aspecific partition $\tilde{\mathcal{P}}_{\min}$. This partition relies on thesame principle as the partition $\mathcal{P}_{\min}$ introduced by Grabisch andSkoda (2012) but restricted to connected coalitions. More precisely, this newpartition $\tilde{\mathcal{P}}_{\min}$ is induced by the deletion of theminimum weight edges in each connected component associated with a coalition.We provide a characterization of the graphs satisfying inheritance of convexityfrom the underlying game to the restricted game associated with$\tilde{\mathcal{P}}_{\min}$.

Inheritance of Convexity for the $\tilde{\mathcal{P}}_{\min}$-Restricted Game