Corpus ID: 210164677

Towards the Small Quasi-Kernel Conjecture

@article{Kostochka2020TowardsTS,
  title={Towards the Small Quasi-Kernel Conjecture},
  author={Alexandr V. Kostochka and Ruth Luo and Songling Shan},
  journal={arXiv: Combinatorics},
  year={2020}
}
Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chv\'atal and Lov\'asz proved that every digraph has a quasi-kernel. P. L. Erd\H{o}s and L. A. Sz\'ekely in 1976 conjectured that if every vertex of $D$ has a positive indegree, then $D$ has a quasi-kernel of size at most $|V|/2$. This conjecture is only confirmed for narrow classes of digraphs… Expand
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