On the number of quasi-kernels in digraphs

A vertex set X of a digraph D = (V, A) is a kernel if X is independent (i.e., all pairs of distinct vertices of X are non-adjacent) and for every v ∈ V − X there exists x ∈ X such that vx ∈ A. A vertex set X of a digraph D = (V, A) is a quasi-kernel if X is independent and for every v ∈ V − X there exist w ∈ V − X, x ∈ X such that either vx ∈ A or vw, wx ∈ A. In 1994, Chvátal and Lovász proved that every digraph has a quasi-kernel. In 1996, Jacob and Meyniel proved that, if