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arXiv · 2307.04112

On the Small Quasi-kernel conjecture

Péter L. Erdős·Ervin Győri·Tamás Róbert Mezei·Nika Salia·Mykhaylo Tyomkyn
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Abstract

An independent vertex subset $S$ of the directed graph $G$ is a kernel if the set of out-neighbors of $S$ is $V(G)\setminus S$. An independent vertex subset $Q$ of $G$ is a quasi-kernel if the union of the first and second out-neighbors contains $V(G)\setminus S$ as a subset. Deciding whether a directed graph has a kernel is an NP-hard problem. In stark contrast, each directed graph has quasi-kernel(s) and one can be found in linear time. In this article, we will survey the results on quasi-kernel and their connection with kernels. We will focus on the small quasi-kernel conjecture which states that if the graph has no vertex of zero in-degree, then there exists a quasi-kernel of size not larger than half of the order of the graph. The paper also contains new proofs and some new results as well.

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Péter L. Erdős, Ervin Győri, Tamás Róbert Mezei, Nika Salia, Mykhaylo Tyomkyn. 2023-07-09. On the Small Quasi-kernel conjecture. https://arxiv.org/abs/2307.04112

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