Large Quasi-kernel Conjecture

Prove that every digraph has a quasi-kernel whose closed out-neighbourhood contains at least half of the vertices.

Background

The closed out-neighbourhood of a set Q is Q together with all vertices that receive an arc from a vertex of Q. The Large Quasi-kernel Conjecture asks whether every digraph, without a source-free hypothesis, has a quasi-kernel whose closed out-neighbourhood contains at least half of the digraph's vertices.

The paper proves that the universal Large Quasi-kernel assertion is equivalent to the universal Small Quasi-kernel assertion, retaining the parameter exactly, but does not establish either conjecture. Thus the large-quasi-kernel statement remains an unresolved target equivalent to the small-quasi-kernel statement.

References

The Small Quasi-kernel Conjecture asserts that every source-free digraph on $n$ vertices has a quasi-kernel of size at most $n/2$, whereas the Large Quasi-kernel Conjecture asks for a quasi-kernel whose closed out-neighbourhood has size at least $n/2$.

— Small and large quasi-kernels: a weighted transfer theorem  (2610.23487 - Ai et al., 16 Oct 2026) in Section 1, Introduction, Conjecture 2 (Large Quasi-kernel Conjecture)