Kernels in planar digraphs

Abstract

A set S of vertices in a digraph D = ( V , A ) is a kernel if S is independent and every vertex in V - S has an out-neighbor in S. We show that there exist O ( n 2 19.1 k + n 4 ) -time and O ( k 36 + 2 19.1 k k 9 + n 2 ) -time algorithms for checking whether a planar digraph D of order n has a kernel with at most k vertices. Moreover, if D has a kernel of size at most k, the algorithms find such a kernel of minimal size.