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Székely László: Towards characterizing 2-crossing critical tanglegrams |
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| Czabarka, Székely and Wagner proved the tanglegram analogue of the Kuratowski theorem: a tanglegram is non-planar precisely when it contains one of two size 4 tanglegrams as subtanglegrams. Call a tanglegram $k$-crossing critical, if it has crossing number at least $k$, but any proper subtanglegram of it has crossing number less than $k$. The tanglegram Kuratowski theorem can be interpreted that the 1-crossing critical tanglegrams are exactly these two size 4 tanglegrams. We showed that the 2-crossing critical tanglegrams have size at most 8, and therefore there are only a finite number of them. We do not know if the number of $k$-crossing critical tanglegrams is finite for $k>2$. We point out analogies and differences between graph minors and subtanglegrams. This is joint work with Czabarka Éva and Alec Helm. |
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