2000 character limit reached
Tight bound for the Erdős-Pósa property of tree minors (2403.06370v2)
Published 11 Mar 2024 in math.CO and cs.DM
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.
- Quickly excluding a forest. Journal of Combinatorial Theory, Series B, 52(2):274–283, 1991.
- Large-treewidth graph decompositions and applications. In Proceedings of the 45th annual ACM Symposium on Theory of Computing, pages 291–300. ACM, 2013.
- Polynomial bounds for the grid-minor theorem. Journal of the ACM, 63(5):40:1–40:65, 2016.
- A tight Erdős-Pósa function for planar minors. Advances in Combinatorics, 10 2019.
- Reinhard Diestel. Graph minors 1: A short proof of the path-width theorem. Combinatorics, Probability and Computing, 4:27–30, 1995.
- On independent circuits contained in a graph. Canadian Journal of Mathematics, 17:347–352, 1965.
- Excluded forest minors and the Erdős–Pósa property. Combinatorics, Probability and Computing, 22(5):700–721, 2013.
- Graph minors. I. Excluding a forest. Journal of Combinatorial Theory, Series B, 35(1):39–61, 1983.
- Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92–114, 1986.