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 $\frac{2}{3}|V(D)|$, 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，拟核是一个独立顶点子集，且每个顶点到该子集的最短路径长度不超过2。Erdős和Szekély于1976年提出的"小拟核猜想"断言：每个无汇有向图都存在一个规模不超过顶点总数某个分数比例的拟核。该猜想在1/2比例下更为精确，但即便允许更大比例，目前也仅能证明少数图类满足此性质。本文聚焦于分割图中的小拟核问题。由于该图族曾被用于否定关于存在两个不相交拟核的强化猜想，其在相关研究中扮演着特殊角色。本文证明：每个无汇分割有向图$D$均存在规模不超过$\frac{2}{3}|V(D)|$的拟核；当图为完全分割有向图的定向时，该规模甚至不超过2。同时证明：计算分割有向图的最小拟核尺寸是W[2]-困难的。