Algorithmic Aspects of Small Quasi-Kernels

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that every vertex can reach some vertex in that subset via a directed path of length at most two. Whereas Chvátal and Lovász proved in 1974 that every digraph has a quasi-kernel, very little is known so far about the complexity of computing small quasi-kernels. In 1976, Erdős and Székely conjectured that every sink-free digraph has a quasi-kernel containing at most half of the vertices. Obviously, if a digraph has two disjoint quasi-kernels then it has such a quasi-kernel and in 2001, Gutin, Koh, Tay and Yeo conjectured that every sink-free digraph has two disjoint quasi-kernels. Yet, they constructed in 2004 a counterexample, thereby disproving this stronger conjecture. We shall show that not only do sink-free digraphs occasionally fail to contain two disjoint quasi-kernels, but it is computationally hard to distinguish those that do from those that do not. We also prove that the problem of computing a smallest quasi-kernel is computationally hard, even for restricted classes of acyclic digraphs and for orientations of split graphs. Finally, we observe that this latter problem is polynomial-time solvable for graphs with bounded treewidth and identify a class of graphs with unbounded treewidth for which the problem is also polynomial-time solvable, namely orientations of complete split graphs.