The Chv\'atal-Lov\'asz theorem from 1974 establishes in every finite digraph $G$ the existence of a quasi-kernel, i.e., an independent $2$-out-dominating vertex set. In the same spirit, the Small Quasi-kernel Conjecture, proposed by Erd\H{o}s and Sz\'ekely in 1976, asserts the existence of a quasi-kernel of order at most $|V(G)|/2$ if $G$ does not have sources. Despite repeated efforts, the conjecture remains wide open. This work contains a number of new results towards the conjecture. In our main contribution we resolve the conjecture for all directed graphs without sources containing a kernel in the second out-neighborhood of a quasi-kernel. Furthermore, we provide a novel strongly connected example demonstrating the asymptotic sharpness of the conjecture. Additionally, we resolve the conjecture in a strong form for all directed unicyclic graphs.

翻译：1974年Chvátal-Lovász定理证明每个有限有向图$G$中都存在一个拟核，即一个独立且2-外支配的顶点集。与此一脉相承，Erdős和Székely于1976年提出的小拟核猜想断言：若$G$无源点，则存在阶数不超过$|V(G)|/2$的拟核。尽管历经多次努力，该猜想仍悬而未决。本文就这一猜想取得一系列新成果。我们的主要贡献在于：解决了所有不含源点、且其拟核的第二外邻域中包含核的有向图的猜想。此外，我们构造了一个新颖的强连通例子，表明该猜想的渐近最优性。最后，我们以强形式解决了所有有向单圈图的猜想。