Localized Erdős-Pósa Property for Subdivisions

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 $\mathrm{\mid }X\mathrm{\mid }\le f(k)$. 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 $m\ge 1$ 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 $\mathrm{\mid }X\mathrm{\mid }\le 2^{f(k)}mk+k(m-n)$.