Feedback vertex sets of digraphs with bounded maximum degree
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909938085789696 |
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| author | Ai, Jiangdong Gutin, Gregory Liu, Xiangzhou Yeo, Anders Zhou, Yacong |
| author_facet | Ai, Jiangdong Gutin, Gregory Liu, Xiangzhou Yeo, Anders Zhou, Yacong |
| contents | A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$ makes it acyclic. Let $D$ be a digraph with $n$ vertices and maximum degree $Δ$. We prove the following bounds. If $D$ is an oriented graph, then $fvs(D)\leq \frac{3n}{7}$ when $Δ\le 4$ and $fvs(D)\leq \frac{n}{2}$ when $Δ\le 5$. If $D$ is a connected digraph, $Δ\le 4$ and $D$ is not obtained from an odd undirected cycle by replacing every edge with the pair of opposite arcs with the same endvertices, then $fvs(D)\leq \frac{n}{2}$. If $D$ is an arbitrary digraph with $Δ\le 5$ then $fvs(D)\leq \frac{2n}{3}.$ Note that all the above bounds are tight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01676 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Feedback vertex sets of digraphs with bounded maximum degree Ai, Jiangdong Gutin, Gregory Liu, Xiangzhou Yeo, Anders Zhou, Yacong Combinatorics A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$ makes it acyclic. Let $D$ be a digraph with $n$ vertices and maximum degree $Δ$. We prove the following bounds. If $D$ is an oriented graph, then $fvs(D)\leq \frac{3n}{7}$ when $Δ\le 4$ and $fvs(D)\leq \frac{n}{2}$ when $Δ\le 5$. If $D$ is a connected digraph, $Δ\le 4$ and $D$ is not obtained from an odd undirected cycle by replacing every edge with the pair of opposite arcs with the same endvertices, then $fvs(D)\leq \frac{n}{2}$. If $D$ is an arbitrary digraph with $Δ\le 5$ then $fvs(D)\leq \frac{2n}{3}.$ Note that all the above bounds are tight. |
| title | Feedback vertex sets of digraphs with bounded maximum degree |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.01676 |