  A tanglegram consists of two rooted binary plane trees with the same number
of leaves and a perfect matching between the two leaf sets. Tanglegrams are
drawn with the leaves on two parallel lines, the trees on either side of the
strip created by these lines, and the perfect matching inside the strip. If
this can be done without any edges crossing, a tanglegram is called planar. We
show that every non-planar tanglegram contains one of two non-planar 4-leaf
tanglegrams as induced subtanglegram, which parallels Kuratowski's Theorem.
