COMPLEXITY OF INHERITANCE OF F-CONVEXITY FOR RESTRICTED GAMES INDUCED BY MINIMUM PARTITIONS

Let G = (N,E,w) be a weighted communication graph. For any subset A subset of N, we delete all minimum-weight edges in the subgraph induced by A. The connected components of the resultant subgraph constitute the partition P-min(A) of A. Then, for every cooperative game (N, v), the P-min-restricted game (N,(v) over bar) is defined by (v) over bar (A) = Sigma(F epsilon Pmin)(A) subset of (F) for all A subset of N. We prove that we can decide in polynomial time if there is inheritance of F-convexity, i.e., if for every F-convex game the Pmin-restricted game is F-convex, where F-convexity is obtained by restricting convexity to connected subsets. This implies that we can also decide in polynomial time for any unweighted graph if there is inheritance of convexity for Myerson's graph-restricted game.