Abstract

The Directed Grid Theorem, stating that there is a function $f$ such that a directed graph of directed treewidth at least $f(k)$ contains a directed grid of size at least $k$ as a butterfly minor, after being a conjecture for nearly 20 years, was proved in 2015 by Kawarabayashi and Kreutzer. However, the function $f$ obtained in the proof is very fast growing. In this work, we show that if one relaxes directed grid to bramble of constant congestion, one can obtain a polynomial bound. More precisely, we show that for every $k \geq 1$ there exists $t = \mathcal{O}(k^{48} \log^{13} k)$ such that every directed graph of directed treewidth at least $t$ contains a bramble of congestion at most 8 and size at least $k$.

Keywords

  1. directed graph
  2. bramble
  3. directed treewidth

MSC codes

  1. 05C20
  2. 05C40
  3. 05C83

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 922 - 938
ISSN (online): 1095-7146

History

Submitted: 5 May 2021
Accepted: 1 February 2022
Published online: 7 April 2022

Keywords

  1. directed graph
  2. bramble
  3. directed treewidth

MSC codes

  1. 05C20
  2. 05C40
  3. 05C83

Authors

Affiliations

Funding Information

Alexander von Humboldt-Stiftung https://doi.org/10.13039/100005156
Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 714704
Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038 : R611450, R611368

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