We study the extended Chinese postman (CP) cooperative game induced by a connected, weighted, undirected graph G. wherein a postman, starting from a post office location, needs to traverse all edges wherein players reside, before returning to the post-office. We characterize the graphs associated with all CP games in which the players on a road pay exactly the cost of the road at each core point, regardless of the number of players residing on the road, the location of the post-office and the edge-weight functions. Here, a road is a maximal path all of whose interior vertices have a degree equal to two in G. For this class of games, the core and nucleolus are Cartesian products of CP games induced by simple cyclic graphs, the core is determined by at most 2n - 1 constraints and the nucleolus can be computed in O(n(2)) time.
Granot, D., Hamers, H., Kuipers, J., & Maschler, M. (2011). On Chinese postman games where residents of each road pay the cost of their road. Games and Economic Behavior, 72(2), 427-438. https://doi.org/10.1016/j.geb.2010.02.002