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arXiv · 2512.21530

Localized Erd\H{o}s-P\'osa Property for Subdivisions

Icey Siyi Ai·Maria Chudnovsky·Julien Codsi
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Abstract

For a graph $H$, we say that $H$ has the Erd\H{o}s-P\'osa property for subdivisions with function $f$, if, for every nonnegative integer $k$ and every graph $G$, either $G$ contains (as a subgraph) $k+1$ pairwise vertex-disjoint subdivisions of $H$ or there exists a set $X\subseteq V(G)$ such that $G\setminus X$ contains no $H$-subdivision and $|X|\leq f(k)$. We show that every connected graph $H$ that has the Erd\H{o}s-P\'osa property for subdivision also satisfies a localized version of the Erd\H{o}s-P\'osa property, as follows. Let $H$ be a connected graph that has the Erd\H{o}s-P\'osa property for subdivisions with function $f$, and let $G$ be a graph that does not contain $k+1$ vertex-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 $|X| \leq 2^{f(k)}mk -k(m-n)$ where $n$ and $m$ are the number of vertices and edges.

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Icey Siyi Ai, Maria Chudnovsky, Julien Codsi. 2025-12-25. Localized Erd\H{o}s-P\'osa Property for Subdivisions. https://arxiv.org/abs/2512.21530

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