On the Small Quasi-kernel conjecture
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The Chvátal-Lovász 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 Erdős and Székely 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.
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