Localized Erdős-Pósa Property for Subdivisions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915694074920960 |
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| author | Ai, Icey Siyi Chudnovsky, Maria Codsi, Julien |
| author_facet | Ai, Icey Siyi Chudnovsky, Maria Codsi, Julien |
| contents | For a graph $H$, we say that $H$ has the Erdős-Pósa property for subdivisions with function $f$, if for every graph $G$, either $G$ contains (as a subgraph) $k+1$ pairwise disjoint subdivisions of $H$ or there exists a set $X\subseteq G$ such that $G\setminus X$ contains no $H$-subdivision and $|X|\leq f(k)$. We show that every $H$ that has the \EP property for subdivision also satisfies a localized version of the \EP property, as follows. Let $H$ be an $n$-vertex graph with $m\geq 1$ edges that has the Erdős-Pósa property for subdivisions with function $f$, and let $G$ be a graph that does not contain $k+1$ disjoint subdivisions of $H$. We demonstrate the existence of a set of at most $k$ vertex disjoint subdivisions of $H$ in $G$ such that in their union, we can find a set $X$ with the property that $G \setminus X$ contains no $H$-subdivision and $|X| \leq 2^{f(k)}mk +k(m-n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21530 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localized Erdős-Pósa Property for Subdivisions Ai, Icey Siyi Chudnovsky, Maria Codsi, Julien Combinatorics 05C70, 05B40 For a graph $H$, we say that $H$ has the Erdős-Pósa property for subdivisions with function $f$, if for every graph $G$, either $G$ contains (as a subgraph) $k+1$ pairwise disjoint subdivisions of $H$ or there exists a set $X\subseteq G$ such that $G\setminus X$ contains no $H$-subdivision and $|X|\leq f(k)$. We show that every $H$ that has the \EP property for subdivision also satisfies a localized version of the \EP property, as follows. Let $H$ be an $n$-vertex graph with $m\geq 1$ edges that has the Erdős-Pósa property for subdivisions with function $f$, and let $G$ be a graph that does not contain $k+1$ disjoint subdivisions of $H$. We demonstrate the existence of a set of at most $k$ vertex disjoint subdivisions of $H$ in $G$ such that in their union, we can find a set $X$ with the property that $G \setminus X$ contains no $H$-subdivision and $|X| \leq 2^{f(k)}mk +k(m-n)$. |
| title | Localized Erdős-Pósa Property for Subdivisions |
| topic | Combinatorics 05C70, 05B40 |
| url | https://arxiv.org/abs/2512.21530 |