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# Improved Bounds for the Excluded-Minor Approximation of Treedepth

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LIPIcs.ESA.2019.34.pdf
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## Cite As

Wojciech Czerwiński, Wojciech Nadara, and Marcin Pilipczuk. Improved Bounds for the Excluded-Minor Approximation of Treedepth. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 34:1-34:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.34

## Abstract

Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant C such that for every integers a,b >= 2 and a graph G, if the treedepth of G is at least Cab log a, then the treewidth of G is at least a or G contains a subcubic (i.e., of maximum degree at most 3) tree of treedepth at least b as a subgraph. As a direct corollary, we obtain that every graph of treedepth Omega(k^3 log k) is either of treewidth at least k, contains a subdivision of full binary tree of depth k, or contains a path of length 2^k. This improves the bound of Omega(k^5 log^2 k) of Kawarabayashi and Rossman [SODA 2018]. We also show an application for approximation algorithms of treedepth: given a graph G of treedepth k and treewidth t, one can in polynomial time compute a treedepth decomposition of G of width O(kt log^{3/2} t). This improves upon a bound of O(kt^2 log t) stemming from a tradeoff between known results. The main technical ingredient in our result is a proof that every tree of treedepth d contains a subcubic subtree of treedepth at least d * log_3 ((1+sqrt{5})/2).

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph theory
##### Keywords
• treedepth
• excluded minor

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## References

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