In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The "small quasi-kernel conjecture," proposed by Erd\H{o}s and Sz\'ekely in 1976, postulates that every sink-free digraph has a quasi-kernel whose size is within a fraction of the total number of vertices. The conjecture is even more precise with a $1/2$ ratio, but even with larger ratio, this property is known to hold only for few classes of graphs. The focus here is on small quasi-kernels in split graphs. This family of graphs has played a special role in the study of the conjecture since it was used to disprove a strengthening that postulated the existence of two disjoint quasi-kernels. The paper proves that every sink-free split digraph $D$ has a quasi-kernel of size at most $\frac{3}{4}|V(D)|$, and even of size at most two when the graph is an orientation of a complete split graph. It is also shown that computing a quasi-kernel of minimal size in a split digraph is W[2]-hard.

翻译：在有向图中，拟核是一类独立顶点子集，满足从任意顶点到该子集的最短路径长度不超过2。由Erdős与Székely于1976年提出的"小拟核猜想"断言：每个无汇有向图均存在大小不超过顶点总数某一比例的拟核。该猜想在1/2比例下更为精确，但即便放宽比例条件，目前仅对少数图类证实成立。本文聚焦分裂图中的小拟核问题。这类图在猜想研究中具有特殊地位——因其曾被用于反驳一个关于存在两个不相交拟核的加强版本。本文证明：每个无汇分裂有向图$D$均存在大小不超过$\frac{3}{4}|V(D)|$的拟核，且当该图是完全分裂有向图的定向图时，拟核大小可降至不超过2。同时证明，在分裂有向图中计算最小拟核属于W[2]-困难问题。