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Tight bound for the Erdős–Pósa property of tree minors

Part of: Graph theory

Published online by Cambridge University Press:  11 December 2024

Vida Dujmović ,
Gwenaël Joret ,
Piotr Micek  and
Pat Morin
Vida Dujmović
Affiliation:
School of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada
Gwenaël Joret*
Affiliation:
Computer Science Department, Université libre de Bruxelles, Brussels, Belgium
Piotr Micek
Affiliation:
Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Pat Morin
Affiliation:
School of Computer Science, Carleton University, Ottawa, Canada
*
Corresponding author: Gwenaël Joret; Email: gwenael.joret@ulb.be
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Abstract

Let $T$ be a tree on $t$ vertices. We prove that for every positive integer $k$ and every graph $G$, either $G$ contains $k$ pairwise vertex-disjoint subgraphs each having a $T$ minor, or there exists a set $X$ of at most $t(k-1)$ vertices of $G$ such that $G-X$ has no $T$ minor. The bound on the size of $X$ is best possible and improves on an earlier $f(t)k$ bound proved by Fiorini, Joret, and Wood (2013) with some fast-growing function $f(t)$. Moreover, our proof is short and simple.

Keywords

Graph minorspathwidthErdős–Pósa property

MSC classification

Primary: 05C83: Graph minors

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 34 , Issue 2 , March 2025 , pp. 321 - 325
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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