2 Game Theory Foundations and Digraph Kernels

Known facts: the decision problem of the existence of a kernel in a digraph G = (V; E) is NP-complete; if all of the cycles of G have even length, then G has a kernel; and the question of the number of kernels is #P-complete even for this restricted class of digraphs. We construct game theory tools, of independent interest, concerning strategies in the presence of draw positions, to show, in the opposite direction, how to partition V , in linear time, into 3 subsets S 1 ; S 2 ; S 3 , such that S 1 lies in all the kernels; S 2 lies in the complements of all the kernels; and on S 3 the kernels may be nonunique. Thus, in particular, digraphs with a \large" number of kernels are those in which S 3 is \large"; possibly S 1 = S 2 = ;. We also show that G can be \decomposed", in linear time, into two induced subgraphs G 1 , with vertex-set S 1 S 2 , which has a unique kernel; and G 2 , with vertex-set S 3 , such that any kernel K of G is the union of the kernel of G 1 and a kernel of G 2. In particular, G has no kernel if and only if G 2 has none.