Degenerate Vertex Cuts in Sparse Graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911354730840064 |
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| author | Hartel, Thilo Rauch, Johannes Rautenbach, Dieter |
| author_facet | Hartel, Thilo Rauch, Johannes Rautenbach, Dieter |
| contents | For a non-negative integer $k$, a vertex cut in a graph is $k$-degenerate if it induces a $k$-degenerate subgraph. We show that a graph of order $n$ at least $2k+2$ without a $k$-degenerate cut has the size at least $\frac{1}{2}\left(k+Ω\left(\sqrt{k}\right)\right)n$ and that a graph of order $n$ at least $5$ without a $2$-degenerate cut has the size at least $\frac{27n-35}{10}$. For $k\geq 2$, we show that a connected graph $G$ of order $n$ at least $k+6$ and size $m$ at most $\frac{k+3}{2}n+\frac{k-1}{2}$ has a minimum $k$-degenerate cut. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Degenerate Vertex Cuts in Sparse Graphs Hartel, Thilo Rauch, Johannes Rautenbach, Dieter Combinatorics For a non-negative integer $k$, a vertex cut in a graph is $k$-degenerate if it induces a $k$-degenerate subgraph. We show that a graph of order $n$ at least $2k+2$ without a $k$-degenerate cut has the size at least $\frac{1}{2}\left(k+Ω\left(\sqrt{k}\right)\right)n$ and that a graph of order $n$ at least $5$ without a $2$-degenerate cut has the size at least $\frac{27n-35}{10}$. For $k\geq 2$, we show that a connected graph $G$ of order $n$ at least $k+6$ and size $m$ at most $\frac{k+3}{2}n+\frac{k-1}{2}$ has a minimum $k$-degenerate cut. |
| title | Degenerate Vertex Cuts in Sparse Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.21298 |