Algorithmic aspects of quasi-kernels
arXiv:2107.03793
Abstract
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 set 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 finding small quasi-kernels. In 1976 Erdős and Székely conjectured that every sink-free digraph has a quasi-kernel of size at most . Obviously, if has two disjoint quasi-kernels then it has a quasi-kernel of size at most , 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 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 small quasi-kernel is polynomial time solvable for orientations of trees but is computationally hard in most other cases (and in particular for restricted acyclic digraphs).