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The grid-minor theorem revisited
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abstract
We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of $H\boxtimes K_c$. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a fixed graph as a minor.
Forward citations
Cited by 2 Pith papers
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Short Paths in the Planar Graph Product Structure Theorem
Every n-vertex planar graph is contained in H ⊠ P ⊠ K_c for some planar H of treewidth 3 and a path P of length O((tw(G)+1)^(1-ε) n^ε).
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Excluding a rectangular grid
A new parameter family, k-treedepth, is characterized by excluded minors T□P_l for all k-vertex trees T, unifying treedepth, the ladder theorem, and the Grid-Minor Theorem.
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