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

TL;DR

This paper proves that graphs with the Erdős-Pósa property for subdivisions also satisfy a localized version, providing bounds on the size of vertex sets needed to eliminate all subdivisions.

Contribution

It establishes a localized Erdős-Pósa property for subdivisions of graphs, extending previous global results with explicit bounds based on the original Erdős-Pósa function.

Findings

Existence of a bounded set of subdivisions covering all subdivisions in the graph.

Explicit bounds on the size of the vertex set needed to eliminate all subdivisions.

Extension of Erdős-Pósa property to a localized context for subdivisions.

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 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 Erd\H{o}s-P\'osa 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 $|X| \leq 2^{f(k)}mk +k(m-n)$.