A tanglegram Kuratowski theorem

TL;DR

This paper establishes a Kuratowski-type theorem for tanglegrams, characterizing non-planar tanglegrams by the presence of specific minimal non-planar configurations, thus advancing the understanding of their planarity properties.

Contribution

It proves that non-planar tanglegrams necessarily contain one of two minimal non-planar 4-leaf tanglegrams, providing a structural characterization similar to Kuratowski's theorem.

Findings

Non-planar tanglegrams contain specific minimal non-planar configurations.

A characterization of tanglegram planarity analogous to Kuratowski's theorem.

Identification of two minimal non-planar 4-leaf tanglegrams.

Abstract

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.