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

We consider restricted games 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 subcomponents corresponding to a minimum partition. This minimum partition $\mathcal{P}_{\min}$ is induced by the deletion of the minimum weight edges. We provide a characterization of the graphs satisfying inheritance of convexity from the underlying game to the restricted game associated with Pmin. Moreover, we prove that these graphs can be recognized in polynomial time.