Recent problems and results about kernels in directed graphs

Let G be a directed graph. Its vertex-set will be denoted by X, and its arcs (or ‘directed edges’) are a subset of the Cartesian product X x X. A kernel of G is a subset S of X which is ‘stable’ (independent, i.e.: a vertex in S has no successor in S) and ‘absorbant’ (dominating, i.e. a vertex not in S has a successor in S). This concept has found many applications, for instance in cooperative n-person games, in Nim-type games (cf. [l]), in logic (cf. [2]), etc. So, the main question is: Which structural properties of a graph imply the existence of a kernel? By ‘subgraph’, we shall always mean ‘induced subgraph’. A graph G whose all subgraphs have kernels is called kernel-perfect. Otherwise, G is kernel-imperfect. The classical results (see [l]) are: (1) A symmetric graph is kernel-perfect (trivial); (2) A transitive graph is kernel-perfect, and all kernels have the same cardinal@ (K&rig); (3) A graph without circuits is kernel-perfect, and its kernel is unique (von Neumann); (4) A graph without odd circuits is kernel-perfect (Richardson). Many extensions of Richardson’s Theorem have been found in the last ten years. An easy one is: