# 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 in V − X there exists x in X such that vx in A. A vertex set

X of a digraph D = (V,A) is a quasi-kernel if X is independent and

for every v in V − X there exist w in V − X, x in X such that either

vx in A or vw,wx in A: In 1994, Chvatal and Lovasz 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. We conjecture that every digraph with no sink

has a pair of disjoint quasi-kernels and provide some support to this

conjecture.

*BRICS Report Series*,

*8*(7). https://doi.org/10.7146/brics.v8i7.20461

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