Quasi-kernels in split graphs

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 & odblac;s and Sz & eacute;kely 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 23 |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. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.