Abstract:A quasi-kernel in a digraph is an independent set of vertices such that any vertex in the digraph can reach some vertex in the set via a directed path of length at most two. Chvátal and Lovász proved that every digraph has a quasi-kernel. Recently, Gutin et al. raised the question of which digraphs have a pair of disjoint quasi-kernels. Clearly, a digraph has a pair of disjoint quasi-kernels cannot contain sinks, that is, vertices of outdegree zero, as each such vertex is necessarily included in a quasi-kernel. However, there exist digraphs which contain neither sinks nor a pair of disjoint quasi-kernels. Thus, containing no sinks is not sufficient in general for a digraph to have a pair of disjoint quasi-kernels. In contrast, we prove that, for several classes of digraphs, the condition of containing no sinks guarantees the existence of a pair of disjoint quasi-kernels. The classes contain semicomplete multipartite, quasi-transitive, and locally semicomplete digraphs.

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