On the number of quasi-kernels in digraphs

Abstract

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 1974, Chvátal and Lovász proved that every digraph has a quasi-kernel. In 1996, Jacob and Meyniel proved that if a digraph D has no kernel, then D contains at least three quasi-kernels. We characterize digraphs with exactly one and two quasi-kernels, and, thus, provide necessary and sufficient conditions for a digraph to have at least three quasi-kernels. In particular, we prove that every strong digraph of order at least three, which is not a 4-cycle, has at least three quasi-kernels.