Description
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.