Localized Erdős-Pósa Property for Subdivisions

TL;DR

The paper studies a localized version of the Erdős-Pósa property for subdivisions: if an $n$-vertex graph $H$ with $m\ge 1$ edges has the Erdős-Pósa property for subdivisions with bound $f_H(k)$, then any graph $G$ without $k+1$ disjoint $H$-subdivisions admits a collection of at most $k$ disjoint $H$-subdivisions whose union contains a hitting set $X$ for all $H$-subdivisions after removing $X$, with $|X|\le 2^{f_H(k)} m k + k(m-n)$. The proof hinges on a constructive framework using $(k,H)$-hitting triples, a carefully defined scoring function, and Menger's theorem to iteratively localize the obstruction within the union of a small number of subdivisions. This yields a quantitative localized EP bound and connects to known results for subcubic graphs, highlighting the structural interplay between packing and localized covering in subdivisions. The approach generalizes Erdős-Pósa phenomena to a locality-constrained setting, with potential implications for graph structure theory and algorithmic applications where localized deletions bound subdivision obstructions.

Abstract

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 $|X| \leq 2^{f(k)}mk +k(m-n)$.

Figures (4)

• Figure 1: Tree $T$ for the counter example of Question \ref{['question1']}
• Figure 2: Graph $G$ for the counter example of Question \ref{['question1']}
• Figure 3: Operation for active paths of type (iii).
• Figure 4: Rerouting from separation.

Theorems & Definitions (11)

• Theorem 1: Erdős-Pósa, 1965
• Theorem 2: Robertson, Seymour, 1986
• Definition 1: Subdivisions
• Theorem 3
• Lemma 1: from diestel2005
• Corollary 1
• Theorem 4: Dujmović, Joret, Micek, Morin, 2024 Dujmovi__2024
• Corollary 2
• Theorem 5: Menger's Theorem Menger1927
• proof : Proof of \ref{['general:main']}