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arXiv · 1807.04969

A tight Erdős-Pósa function for planar minors

Wouter Cames van Batenburg·Tony Huynh·Gwenaël Joret·Jean-Florent Raymond
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Abstract

Let $H$ be a planar graph. By a classical result of Robertson and Seymour, there is a function $f:\mathbb{N} \to \mathbb{R}$ such that for all $k \in \mathbb{N}$ and all graphs $G$, either $G$ contains $k$ vertex-disjoint subgraphs each containing $H$ as a minor, or there is a subset $X$ of at most $f(k)$ vertices such that $G-X$ has no $H$-minor. We prove that this remains true with $f(k) = c k \log k$ for some constant $c=c(H)$. This bound is best possible, up to the value of $c$, and improves upon a recent result of Chekuri and Chuzhoy [STOC 2013], who established this with $f(k) = c k \log^d k$ for some universal constant $d$. The proof is constructive and yields a polynomial-time $O(\log \mathsf{OPT})$-approximation algorithm for packing subgraphs containing an $H$-minor.

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Wouter Cames van Batenburg, Tony Huynh, Gwenaël Joret, Jean-Florent Raymond. 2019-10-23. A tight Erdős-Pósa function for planar minors. https://doi.org/10.19086/aic.10807

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