Localized Erdős-Pósa Property for Subdivisions

Abstract

For a graph $H$, we say that $H$ has the Erdős-Pósa 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 $|X|\le f(k)$. We show that every connected graph $H$ that has the Erdős-Pósa property for subdivision also satisfies a localized version of the Erdős-Pósa property, as follows. Let $H$ be a connected graph that has the Erdős-Pósa 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 $|X|\le 2^{f(k)}mk-k(m-n)$ where $n$ and $m$ are the number of vertices and edges.