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Improved bounds for the excluded-minor approximation of treedepth (1904.13077v2)

Published 30 Apr 2019 in cs.DM, cs.DS, and math.CO

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 positive integers $a,b$ and a graph $G$, if the treedepth of $G$ is at least $Cab$, 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 $\Omega(k3)$ is either of treewidth at least $k$, contains a subdivision of full binary tree of depth $k$, or contains a path of length $2k$. This improves the bound of $\Omega(k5 \log2 k)$ of Kawarabayashi and Rossman [SODA 2018]. We also show an application of our techniques 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 $\mathcal{O}(kt \log{3/2} t)$. This improves upon a bound of $\mathcal{O}(kt2 \log t)$ 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 $d \cdot \log_3 ((1+\sqrt{5})/2)$.

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Authors (3)
  1. Wojciech CzerwiƄski (35 papers)
  2. Wojciech Nadara (16 papers)
  3. Marcin Pilipczuk (115 papers)
Citations (23)

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