A tight Erdős-Pósa function for planar minors
Advances in Combinatronics (2019)
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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.
Publié le :
@article{ADVC_2019_a3,
author = {Wouter Cames van Batenburg and Tony Huynh and Gwena\"el Joret and Jean-Florent Raymond},
title = {A tight {Erd\H{o}s-P\'osa} function for planar minors},
journal = {Advances in Combinatronics},
publisher = {mathdoc},
year = {2019},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ADVC_2019_a3/}
}
Wouter Cames van Batenburg; Tony Huynh; Gwenaël Joret; Jean-Florent Raymond. A tight Erdős-Pósa function for planar minors. Advances in Combinatronics (2019). http://geodesic.mathdoc.fr/item/ADVC_2019_a3/