Small Quasi-kernel Conjecture

Prove that every source-free digraph on n vertices has a quasi-kernel of size at most n/2.

Background

A quasi-kernel is an independent vertex set from which every vertex is reachable by a directed path of length at most two. The Small Quasi-kernel Conjecture concerns source-free digraphs, meaning digraphs with no vertex of indegree zero, and asks for a quasi-kernel containing at most half of all vertices.

The paper establishes a weighted transfer theorem and proves that the universal small and large quasi-kernel assertions are equivalent, but it does not prove the Small Quasi-kernel Conjecture itself. The directed 4-cycle demonstrates that the constant one-half would be best possible if the conjecture holds.

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 1 (Small Quasi-kernel Conjecture)