Abstract
A weighted minimum colouring (WMC) game is induced by an undirected graph and a positive weight vector on its vertices. The value of a coalition in a WMC game is determined by the weighted chromatic number of its induced subgraph. A graph G is said to be globally (respectively, locally) WMC totally balanced, submodular, or PMAS-admissible, if for all positive integer weight vectors (respectively, for at least one positive integer weight vector), the corresponding WMC game is totally balanced, submodular or admits a population monotonic allocation scheme (PMAS). We show that a graph G is globally WMC totally balanced if and only if it is perfect, whereas any graph G is locally WMC totally balanced. Furthermore, G is globally (respectively, locally) WMC submodular if and only if it is complete multipartite (respectively, \((2K_2,P_4)\)-free). Finally, we show that G is globally PMAS-admissible if and only if it is \((2K_2,P_4)\)-free, and we provide a partial characterisation of locally PMAS-admissible graphs.
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Notes
Highway games are cooperative cost allocation games in which the cost to be allocated is associated with the construction of a highway network.
Quasi threshold graphs are also called comparability graphs Wölk (1965) or trivially perfect graphs.
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Trine Platz gratefully acknowledges financial support from the Independent Research Fund Denmark \(\mid \) Social Sciences (Grant ID: DFF-6109-000132)
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Hamers, H., Horozoglu, N., Norde, H. et al. On the properties of weighted minimum colouring games. Ann Oper Res 318, 963–983 (2022). https://doi.org/10.1007/s10479-021-04374-9
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DOI: https://doi.org/10.1007/s10479-021-04374-9