Combinatorial game theory foundations applied to digraph kernels

Abstract

Known complexity 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. In the opposite direction, we construct game theory tools, of independent interest, concerning strategies in the presence of draw positions, to show how to partition V, in O(|E|) time, into 3 subsets S1, S2, S3, such that S1 lies in all the kernels; S2 lies in the complements of all the kernels; and on S3 the kernels may be nonunique. Thus, in particular, digraphs with a "large" number of kernels are those in which S3 is "large"; possibly S 1 = S2 = ∅. We also show that G can be decomposed, in O(|E|) time, into two induced subgraphs G1, with vertex-set S 1 ∪ S2, which has a unique kernel; and G2, with vertex-set S3, such that any kernel K of G is the union of the kernel of G1 and a kernel of G2. In particular, G has no kernel if and only if G2 has none. Our results hold even for some classes of infinite digraphs.