Perfect graphs, kernels, and cores of cooperative games

A kernel of a directed graph D is defined as an independent set which is reachable from each outside vertex by an arc. A graph G is called kernel-solvable if an orientation D of G has a kernel whenever each clique of G has a kernel in D. The notion of kernel-solvability has important applications in combinatorics, list coloring, and game theory. It turns out that kernel-solvability is equivalent to perfectness, as it was conjectured by Berge and Duchet in 1983. These and other kernel-related results are the subject of the present survey. Many of these results are independent of the strong perfect graph conjecture, yet, the recent proof of this conjecture and the efficient recognition of perfect graphs have several important implications, in particular in game theory, which are also included here. © 2006 Elsevier B.V. All rights reserved.