Every k-tangle in a graph is the lift of a k-tangle in a topological minor of size bounded by a function of k, reducing the vertex-set induction problem to bounded-size graphs.
K_6 minors in 6-connected graphs of bounded tree-width
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abstract
We prove that every sufficiently big 6-connected graph of bounded tree-width either has a K_6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently big 6-connected graphs. Jorgensen conjectured that it holds for all 6-connected graphs.
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On vertex sets inducing tangles
Every k-tangle in a graph is the lift of a k-tangle in a topological minor of size bounded by a function of k, reducing the vertex-set induction problem to bounded-size graphs.