Abstract
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 sub-components 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 \(\mathcal {P}_{\min }\). Moreover, we prove that these graphs can be recognized in polynomial time.
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Notes
\(\mathcal {F}\) is weakly union-closed if \(A \cup B \in \mathcal {F}\) for all A, \(B \in \mathcal {F}\) such that \(A \cap B \not = \emptyset \) (Faigle et al. 2010). Weakly union-closed families were introduced and analysed by Algaba (1998) (see also Algaba et al. 2000) and called union stable systems.
A vertex in a graph is an articulation point if its removal disconnects the graph.
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Skoda, A. Inheritance of convexity for the \(\mathcal {P}_{\min }\)-restricted game. Math Meth Oper Res 93, 1–32 (2021). https://doi.org/10.1007/s00186-020-00728-4
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DOI: https://doi.org/10.1007/s00186-020-00728-4