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- W2066666508 abstract "Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $elltimesell$ grid minor is exponential in $ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A emph{grid-like-minor of order} $ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{ell}$-minor. For example, the rows and columns of the $elltimesell$ grid are a grid-like-minor of order $ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $cell^4sqrt{logell}$ has a grid-like-minor of order $ell$. As an application of this result, we prove that the cartesian product $Gsquare K_2$ contains a $K_{ell}$-minor whenever $G$ has treewidth at least $cell^4sqrt{logell}$." @default.
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- W2066666508 date "2012-04-01" @default.
- W2066666508 modified "2023-09-29" @default.
- W2066666508 title "Polynomial treewidth forces a large grid-like-minor" @default.
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- W2066666508 doi "https://doi.org/10.1016/j.ejc.2011.09.004" @default.
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