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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.05703 |
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| _version_ | 1866914030922235904 |
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| author | Gutiérrez, Juan |
| author_facet | Gutiérrez, Juan |
| contents | A dual version of a conjecture by Woodall asserts that, in a planar digraph, the length of a shortest dicycle equals the maximum number of pairwise disjoint feedback arc sets. We verify this conjecture for the case where the underlying graph is a 3-tree or a partial 3-tree with girth $3$. Additionally, we show that every 3-tree has a feedback arc set of size at most~$m/3-1$, where~$m$ is the number of arcs of the digraph, and this bound is tight. We further establish an upper bound on the size of a minimum feedback arc set in $k$-trees. Finally, we discuss some open problems and conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards a Dual Version of Woodall's Conjecture for Partial 3-Trees Gutiérrez, Juan Combinatorics Discrete Mathematics 05C20 G.2.2 A dual version of a conjecture by Woodall asserts that, in a planar digraph, the length of a shortest dicycle equals the maximum number of pairwise disjoint feedback arc sets. We verify this conjecture for the case where the underlying graph is a 3-tree or a partial 3-tree with girth $3$. Additionally, we show that every 3-tree has a feedback arc set of size at most~$m/3-1$, where~$m$ is the number of arcs of the digraph, and this bound is tight. We further establish an upper bound on the size of a minimum feedback arc set in $k$-trees. Finally, we discuss some open problems and conjectures. |
| title | Towards a Dual Version of Woodall's Conjecture for Partial 3-Trees |
| topic | Combinatorics Discrete Mathematics 05C20 G.2.2 |
| url | https://arxiv.org/abs/2408.05703 |