Quasi-kernels in Hereditary Classes and Applications to Break

Recently, Nguyen, Seymour and Scott verified the small quasi-kernel conjecture for split digraphs, and initiated the study of quasi-kernels in break digraphs. Following their research, we introduce a weighted half-neighborhood property for hereditary classes of oriented graphs and show that it gives a $2n\mathrm{/}3$ bound of small quasi-kernel for break digraphs. We also record two stronger $n\mathrm{/}2$ results for special classes of break digraphs. Finally, using the same framework we also prove that every digraph on $n$ vertices has a quasi-kernel $Q$ with $\mathrm{\mid }{N}_D^{+}[Q]\mathrm{\mid }\ge \sqrt{n}$.