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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.04112 |
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| _version_ | 1866916264167866368 |
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| author | Erdős, Péter L. Győri, Ervin Mezei, Tamás Róbert Salia, Nika Tyomkyn, Mykhaylo |
| author_facet | Erdős, Péter L. Győri, Ervin Mezei, Tamás Róbert Salia, Nika Tyomkyn, Mykhaylo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04112 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Small Quasi-kernel conjecture Erdős, Péter L. Győri, Ervin Mezei, Tamás Róbert Salia, Nika Tyomkyn, Mykhaylo Combinatorics Discrete Mathematics 05C20 05C69 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. |
| title | On the Small Quasi-kernel conjecture |
| topic | Combinatorics Discrete Mathematics 05C20 05C69 |
| url | https://arxiv.org/abs/2307.04112 |