Polynomial Planar Directed Grid Theorem

Hatzel, Meike; Kawarabayashi, Ken-ichi; Kreutzer, Stephan

doi:10.1137/1.9781611975482.89

Cited by 6 publications

(7 citation statements)

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“…So far, similar bounds were known only for planar graphs, where Hatzel, Kawarabayashi, and Kreutzer showed a polynomial bound (with degree 6 of the polynomial) for the Directed Grid Theorem [7]. Decreasing the congestion in Theorem 1.2, ideally to 2, even at the cost of higher polynomial dependency of t and k, remains an interesting open problem.…”

Section: Introductionmentioning

confidence: 84%

“…We also use their Lovász Local Lemma-based argument to find a large independent set in a multipartite graph of low degeneracy. Similarly as in the proof of Directed Grid Theorem [12] and in its planar variant [7], we start from the notion of a path system and its existence (with appropriate parameters) in graphs of high directed treewidth. Finally, from our recent proof of half-and quarter-integral directed Erdős-Pósa property [15,14], we reuse their partitioning lemma, allowing us to find a large number of closed walks with small congestion.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, from our recent proof of half-and quarter-integral directed Erdős-Pósa property [15,14], we reuse their partitioning lemma, allowing us to find a large number of closed walks with small congestion. On top of the above, compared to [7] and [15,14], the proof of Theorem 1.2 offers a much more elaborate analysis of the studied path system, allowing us to find the desired bramble.…”

Section: Introductionmentioning

confidence: 99%

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Constant congestion brambles in directed graphs

Masařík¹,

Pilipczuk²,

Rzążewski³

et al. 2021

Preprint

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The Directed Grid Theorem, stating that there is a function f such that a directed graphs 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, has been proven 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 ≥ 1 there exists t = 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.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constant congestion brambles in directed graphs

Masařík¹,

Pilipczuk²,

Rzążewski³

et al. 2021

Preprint

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“…The proof of Lemma 13 is inspired by the proof of Lemma 5.4 in [12]. We could use Lemma 5.4 here as well, but its proof, unfortunately, contains errors.…”

Section: The Dense Casementioning

confidence: 99%

“…On the technical side, the proof of Theorem 5 borrows a number of technical tools from the recent work of Hatzel, Kawarabayashi, and Kreutzer that proved polynomial bounds for the directed grid minor theorem in planar graphs [12]. We follow their general approach to obtain a directed treewidth sparsifier [12,Section 5] and modify it in a number of places for our goal. The main novelty comes in different handling of the case when two linkages intersect a lot.…”

Section: Introductionmentioning

confidence: 99%

Packing Directed Cycles Quarter- and Half-Integrally

Masařík¹,

Muzi²,

Pilipczuk³

et al. 2019

Preprint

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The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size O(k log k). After being known for long as Younger's conjecture, a similar statement for directed graphs has been proven in 1996 by Reed, Robertson, Seymour, and Thomas. However, in their proof, the dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing is not elementary.We show that if we compare the size of a minimum feedback vertex set in a directed graph with quarter-integral cycle packing number, we obtain a polynomial bound. More precisely, we show that if in a directed graph G there is no family of k cycles such that every vertex of G is in at most four of the cycles, then there exists a feedback vertex set in G of size O(k 4 ). On the way there we prove a more general result about quarter-integral packing of subgraphs of high directed treewidth: for every pair of positive integers a and b, if a directed graph G has directed treewidth Ω(a 6 b 8 log 2 (ab)), then one can find in G a family of a subgraphs, each of directed treewidth at least b, such that every vertex of G is in at most four subgraphs.

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