Results on the small quasi-kernel conjecture

A quasi-kernel of a digraph D is an independent set Q⊆V(D) such that for every vertex v∈V(D)﹨Q, there exists directed path with one or two arcs from v to u∈Q. In 1974, Chvátal and Lovász proved has quasi-kernel. 1976, Erdős Székely conjectured sink-free D=(V(D),A(D)) size at most |V(D)|/2. this paper, we give new method show the conjecture holds generalization anti-claw-free digraphs. For any one-way split order n, when n≥3, stronger result n+32−n, bound sharp.