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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| Online-Zugang: | https://arxiv.org/abs/2007.12257 |
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| _version_ | 1866913625662291968 |
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| author | Kawarabayashi, Ken-ichi Kreutzer, Stephan Kwon, O-joung Xie, Qiqin |
| author_facet | Kawarabayashi, Ken-ichi Kreutzer, Stephan Kwon, O-joung Xie, Qiqin |
| contents | We prove that there exists a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that every directed graph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a set of at most $f(k)$ vertices meeting all directed odd cycles. We also give a polynomial-time algorithm for fixed $k$ which outputs one of the two outcomes. Using this algorithmic result, we give a polynomial-time algorithm for fixed $k$ to decide whether such $k$ directed odd cycles exist, or there are no $k$ vertex-disjoint directed odd cycles. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_12257 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A half-integral Erdős-Pósa theorem for directed odd cycles Kawarabayashi, Ken-ichi Kreutzer, Stephan Kwon, O-joung Xie, Qiqin Combinatorics We prove that there exists a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that every directed graph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a set of at most $f(k)$ vertices meeting all directed odd cycles. We also give a polynomial-time algorithm for fixed $k$ which outputs one of the two outcomes. Using this algorithmic result, we give a polynomial-time algorithm for fixed $k$ to decide whether such $k$ directed odd cycles exist, or there are no $k$ vertex-disjoint directed odd cycles. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs. |
| title | A half-integral Erdős-Pósa theorem for directed odd cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2007.12257 |