The kk-centrum Chinese Postman delivery problem and a related cost allocation game

We analyze the cost allocation cooperative game, denoted by (N,ck)(N,ck), induced by the kk-centrum Chinese Postman (kk-centrum CP) delivery problem in connected undirected and strongly connected directed graphs. For the undirected case, we show, for example, that for k=1,2k=1,2, (N,ck)(N,ck) is submodular for all graphs having non-negative edge-weights, for all locations of the post-office. For k≥3k≥3, we prove that (N,ck)(N,ck) is submodular for all non-negative edge-weights and for all locations of the post office if and only if the graph is either the cyclic graph on three edges or it is a graph wherein each edge is contained in at most one cycle and these cycles, if any, have only two edges. For the directed graph case we show, for example, that the kk-centrum CP game induced by a strongly connected graph GG is submodular for all k≥2k≥2 if and only if every arc in GG is contained in precisely one directed cycle.